978-3-98547-561-2-volume-3

INTERNATIONAL GONGRESS

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. The main point is that the notion of a finite correspondence for smooth finite type schemes over a field extends to a corresponding notion over a general base-scheme (see

[33, $ 8]

). This gives rise to a theory of motivic cohomology generalizing Voevodsky's definition as

H^{p, q} (X, Z) := {Hom}_{{D M}_{C D} (S)} (M (X), Z_{S} (q) [p])

for

X

smooth over

S

. They show that the assignment

S \mapsto {DM}_{C D} (S)

defines a functor to the category of triangulated tensor categories,

{D M}_{C D} (-) : {S c h}_{B}^{o p} \to {T r}^{\otimes}

, admitting a sixfunctor formalism. There are also Tate twists

M \mapsto M (n)

. This gives a definition of motivic cohomology of an general scheme

Y

which for

Y \in {S m}_{S}

agrees with the definition given above.

They construct an adjunction

ϕ^{*} : S H (Y) ⇄ {DM}_{C D} (Y) : ϕ_{*}

with

ϕ_{*}

playing the role of the Eilenberg-MacLane functor, giving rise to the spectrum

M_{Y} \in S H (Y)

representing

H^{*, *} (Y, Z)

[33, DEfInITION 11.2.17]. They discuss the question of whether

Y \mapsto M Z_{Y}

is cartesian (see [33, CONJECTURE 11.2.22, PROPOSITION 11.4.7]), without reaching a general resolution.

Cisinski-DÃ©glise have a different approach for representing motivic cohomology with

Q

-coefficients, much in the same spirit as Beilinson's construction of universal cohomology using algebraic

K

-theory. Using the spectrum

{KGL}_{S} \in S H (S)

, which represents homotopy invariant algebraic

K

-theory, they use the Adams operations to decompose

{K G L}_{S Q}

into summands,

{K G L}_{S Q} = ⨁_{i} {K G L}_{S}^{(i)}

, with

{K G L}_{S}^{(i)}

representing the

i

th graded piece of

K

-theory for the

γ

-filtration. This gives them a nice commutative monoid object (i.e., commutative ring spectrum)

H_{S}^{E} := {K G L}_{S}^{(0)} \in S H (S)_{Q}

, whose module category they call the category of Beilinson motives over

S

. This construction is cartesian, gives a good theory of motivic cohomology with

Q

-coefficients over a general base-scheme and agrees with

{D M}_{C D} (S)_{Q}

for

S

a uni-branch scheme. See [33, Â§14] for details.

3.2. Spitzweck's motivic cohomology

In [110], Spitzweck constructs a motivic cohomology theory over an arbitrary basescheme. The Bloch cycle complex gives rise to a general version of Bloch's higher Chow groups for finite type schemes over a Dedekind domain, which has nice localization properties (by [25] and [84]), but has poor functoriality and lacks a multiplicative structure. On the other hand, using the Bloch-Kato conjectures, established by Voevodsky et al., the

ℓ

-completed higher Chow groups are recognized as a truncated

ℓ

-adic Ã©tale cohomology, for

ℓ

prime to all residue characteristics. The theorem of Geisser-Levine [52] describes the

p

-completed higher Chow groups in characteristic

p > 0

in terms of logarithmic de RhamWitt sheaves. Finally, there is the good theory with

Q

-coefficients given by Beilinson motivic cohomology of Cisinski-DÃ©glise, as described above.

Each of these three theories, namely the

ℓ

-adic Ã©tale cohomology, the cohomology of the logarithmic de Rham-Witt sheaves, and the rational Beilinson motivic cohomology, has good functoriality and multiplicative properties. Gluing the

ℓ

-adic,

p

-adic, and rational theories together via their respective comparisons with the Bloch cycle complex, Spitzweck constructs a theory with good functoriality and multiplicative properties, and which is described by a presheaf of complexes on smooth schemes over a given Dedekind domain as base-scheme. The corresponding theory agrees with Voevodsky's motivic cohomology for smooth schemes over a perfect field, and is given additively by the hypercohomology of the Bloch complex for smooth schemes over a Dedekind domain (even in mixed characteristic).

Taking the base-scheme to be Spec

Z

, Spitzweck's construction yields a representing object

M Z_{Z}

S H (Z)

and one can thus define absolute motivic cohomology for smooth schemes over a given base-scheme

S

by pulling back

M Z_{Z}

M Z_{S} \in S H (S)

. The resulting motivic cohomology agrees with Voevodsky's for smooth schemes of finite type over a perfect base-field, and with the hypercohomology of the Bloch cycle complex for smooth finite type schemes over a Dedekind domain. This gives rise to a triangulated category of motives

{D M}_{s p} (S)

over a base-scheme

S

, defined as the homotopy category of

M Z_{S^{-}}

modules, and the functor

S \mapsto {D M}_{S p} (S)

inherits a Grothendieck six-functor formalism from that of

S \mapsto S H (S)

3.3. Hoyois' motivic cohomology

Spitzweck's construction gives a solution to the problem of constructing a triangulated category of motives over an arbitrary base, admitting a six-functor formalism and thus yielding a good theory of motivic cohomology. His construction is a bit indirect and it would be nice to have a direct construction of a representing motivic ring spectrum

H Z_{S} \in S H (S)

for each base-scheme

S

, still satisfying the cartesian condition.

Hoyois has constructed such a theory of motivic cohomology over an arbitrary basescheme by using a recent breakthrough in our understanding of the motivic stable homotopy categories

S H (S)

. This is a new construction of

S H (S)

more in line with Voevodsky construction of

D M (k)

. The basic idea is sketched in notes of Voevodsky [126], which were realized in a series of works by Ananyevskiy, Garkusha, Panin, Neshitov [2,4,45-48](authorship in various combinations). Building on these works, Elmanto, Hoyois, Khan, Sosnilo, and Yakerson [36-38] construct an infinity category of framed correspondences, and use the basic program of Voevodsky's construction of

D M (k)

to realize

S H (S)

as arising from presheaves of spectra with framed transfers, just as objects of

D M (k)

arise from presheaves of complexes of sheaves with transfers for finite correspondences. It is not our purpose here to give a detailed discussion of this beautiful topic; we content ourselves with sketching some of the basic principles.

An integral closed subscheme

Z \subset X \times Y

that defines a finite correspondence from

X

Y

can be thought of a special type of a span via the two projections

For

X

and

Y

smooth and finite type over a given base-scheme

S

, a framed correspondence from

X

Y

is also a span,

satisfying certain conditions, together with some additional data (the framing). For simplicity, assume that

X

is connected. The morphism

p

is required to be a finite, flat, local complete intersection (lci) morphism, called a finite syntomic morphism (the terminology was introduced by Mazur). The lci condition means that

p

factors as closed immersion

i : Z \to P

followed by a smooth morphism

f : P \to X

, and the closed subscheme

i (Z)

P

is locally defined by exactly

\dim_{X} P - \dim_{X} Z

equations forming a regular sequence. The morphism

p

factored in this way has a relative cotangent complex

L_{p}

admitting a simple description, namely

L_{p} = [l_{Z} / l_{Z}^{2} \overset{d}{\to} i^{*} Ω_{P / X}]

the conditions on

i

and

p

say that both

ℓ_{Z} / ℓ_{Z}^{2}

and

i^{*} Ω_{P / X}

are locally free coherent sheaves on

Z

rank \dim_{X} P - \dim_{X} Z

and

\dim_{X} P

, respectively. For

p

an lci morphism, the perfect
complex

L_{p}

defines a point

{L_{p}}

in the space

K (Z)

defining the

K

-theory of

Z

; in the case of a finite syntomic morphism, the virtual rank of

{L_{p}}

is zero.

A framing for a syntomic map

p : Z \to X

is a choice of a path

γ : [0, 1] \to K (Z)

connecting

{L_{p}}

with the base-point

0 \in K (Z)

. For a framing to exist, the class

[L_{p}] \in K_{0} (Z)

must be zero, but the choice of

γ

is additional data. The morphism

q : Z \to Y

is arbitrary.

One has the usual notion of a composition of spans:

which preserves the finite syntomic condition. However, one needs a higher categorical structure to take care of associativity constraints. The composition of paths is even trickier, since we are dealing here with actual paths, not paths up to homotopy. In the end, this produces an infinity category Corr

^{f r} ({S m}_{S})

of framed correspondences on smooth

S

-schemes, rather than a category; roughly speaking, the composition is only defined "up to homotopy and coherent higher homotopies."

Via the infinity category

{Corr}^{f r} ({S m}_{S})

, we have the infinity category of framed motivic spaces,

H^{f r} (S)

, this being the infinity category of

A^{1}

-invariant, Nisnevich sheaves of spaces on

C o r r^{f r} ({S m}_{S})

. There is a stable version,

{S H}^{f r} (S)

, an infinite suspension functor

Σ_{f r}^{\infty} : H^{f r} (S) \to {S H}^{f r} (S)

, and an equivalence of infinity categories

γ_{*} : {S H}^{f r} (S) \to S H (S)

, where

S H (S)

is the infinity category version of the triangulated category

S H (S)

, that is, the homotopy category of

S H (S)

SH (S)

. The equivalence

γ_{*}

can be thought of as a version of the construction of infinite loop spaces from Segal's

Γ

-spaces, with a framed correspondence

X \leftarrow Z \to Y

of degree

n

over

X

being viewed as a generalization of the map

[n]_{+} \to [0]_{+}

Γ^{op}

With this background, we can give a rough idea of Hoyois' construction of the spectrum representing motivic cohomology over

S

in [63]. He considers spans

X \overset{p}{\leftarrow} Z \overset{q}{\to} Y

X, Y \in {Sm}_{S}

, with

p : Z \to X

a finite morphism such that

p_{*} O_{Z}

is a locally free

O_{X^{-}}

module; note that this condition is satisfied if

p

is a syntomic morphism, but not conversely. These spans form a category Corr

^{flf} ({S m}_{S})

under span composition ("flf" stands for "finite, locally free") and forgetting the paths

γ

defines a morphism of (infinity) categories

π_{a d} : {Corr}^{f r} ({S m}_{S}) \to {Corr}^{f l f} ({S m}_{S})

Given a commutative monoid

A

, the constant Nisnevich sheaf on

{S m}_{S}

with value

A

extends to a functor

A_{S} : {({C o r r}^{f l f})}^{o p} \to A b

where the pullback from

Y

X

X \overset{p}{\leftarrow} Z \overset{q}{\to} Y

is given by multiplication by

{r n k}_{O_{X}} O_{Z}

X

and

Y

are connected; one extends to general smooth

X

and

Y

by additivity. This gives us the presheaf (of abelian monoids) with framed transfers

A_{S}^{f r} := A_{S} \circ π_{a d}^{o p}

, and the machinery of [36-38] converts this into the motivic spectrum

γ_{*} Σ_{f r}^{\infty} A_{S}^{f r} \in SH (S)

. Hoyois shows [63,

LEMMA 20] that this construction produces a cartesian family, and that taking

A = Z

recovers Spitzweck's family

S \mapsto M Z_{S}

[63, THEOREM 21].

This gives us a conceptually simple construction of a motivic Eilenberg-MacLane spectrum, and the corresponding motivic category

{D M}_{H} (S)

, much in the spirit of Voevodsky original construction of

D M (k)

and the RÃ¶ndigs- stvÃ¦r theorem identifying

D M (k)

with the homotopy category of

EM (Z (0))

-modules.

4. MILNOR-WITT MOTIVIC COHOMOLOGY

The classical Chow group

{C H}^{n} (X)

of codimension

n

algebraic cycles modulo rational equivalence on a smooth variety

X

is part of the motivic cohomology of

X

via the isomorphism

{C H}^{n} (X) = H^{2 n} (X, Z (n))

. Barge and Morel [12] have introduced a refinement of the Chow groups, the Chow-Witt groups, that incorporates information about quadratic forms. Their construction has been embedded in a larger theory of Milnor-Witt motives and Milnor-Witt motivic cohomology, which we describe in this section. The quadratic information given by the Chow-Witt groups, Milnor-Witt motivic cohomology and related theories has proven useful in recent efforts to give quadratic refinements for intersection theory and enumerative geometry; see

[10, 11, 21, 61, 76, 77, 86]

for some examples. We refer the reader to

[8, 31, 39, 92]

for details on the theory described in this section.

4.1. Milnor-Witt $K$ -theory and the Chow-Witt groups

A codimension

n

algebraic cycle

Z := \sum_{i} n_{i} Z_{i}

can be thought of as the set of its generic points

z_{i}

together with the

Z

-valued function

n_{i}

z_{i}

, from which we can write the group

Z^{n} (X)

of codimension

n

algebraic cycles as

Z^{n} (X) = ⨁_{z \in X^{(n)}} Z

where

X^{(n)}

is the set of points

z \in X

with closure

Z := \bar{z} \subset X

of codimension

n

Let

G W (F)

denote the Grothendieck-Witt ring of virtual non-degenerate quadratic forms over

F

and let

W (F) = GW (F) / (H)

where

H

is the hyperbolic form

H (x, y) =

x^{2} - y^{2}

(we assume throughout that the characteristic is

\neq 2

to avoid technical difficulties);

W (F)

is the Witt ring of anisotropic quadratic forms over

F

(see [107]).

One can consider a finite set of codimension

n

points

z_{i} \in X^{(n)}

, together with a collection of classes

{q_{i} \in G W (k (z_{i}))}

; one recovers a

Z

-valued function on

z_{i}

by taking the rank of

q_{i}

. This gives the group

{\tilde{Z}}^{n} (X) := \underset{z \in X^{(n)}}{} G W (k (z))

with rank homomorphism rnk:

{\tilde{Z}}^{n} (X) \to Z^{n} (X)

. In contrast with integer-valued functions, an element

q \in G W (k (z))

does not always extend to all of

\bar{z}

; there is an obstruction given by a certain boundary map

\partial : GW (k (z)) \to ⨁_{w \in \bar{z} \cap X^{(n + 1)}} W (k (w))

This starts to look more like classical homology, in that one should consider

{\tilde{Z}}^{n} (X)

as a group of chains rather than a group of cycles.

This is not enough, as one needs a quadratic refinement for the classical relation given by rational equivalence. The original construction of Barge-Morel defined this relation, but later developments put their construction in a rather more natural form, which we now describe.

We recall that the Milnor

K

-theory ring

K_{*}^{M} (F) := ⨁_{n \geq 0} K_{n}^{M} (F)

of a field

F

is defined as the quotient of the tensor algebra on the abelian group of units

F^{\times}

, modulo the Steinberg relation

The quadratic refinement of

K_{*}^{M} (F)

is the Hopkins-Morel Milnor-Witt

K

-theory of

F

Definition 4.1 (Hopkins-Morel [92, Definition 6.3.1]). Let

F

be a field. The Milnor-Witt

K

-theory of

F, K_{*}^{M W} (F) := ⨁_{n \in Z} K_{n}^{M W} (F)

, is the

Z

-graded associative algebra defined by the following generators and relations.

Generators

(G1) For each

u \in F^{\times}

, we have the generator

[u]

of degree 1 ;

(G2) There is an additional generator

η

of degree -1 .

Relations

(R0)

η \cdot [u] = [u] \cdot η

;

(R1)

[u v] = [u] + [v] + η \cdot [u] \cdot [v]

;

(R2)

[u] \cdot [1 - u] = 0

for

u \in F ∖ {0, 1}

;

(R3) Let

h = (2 + η \cdot [- 1])

. Then

η \cdot h = 0

It follows directly that sending

[u]

{u} \in K_{1}^{M} (F)

and sending

η

to zero defines a surjective graded algebra homomorphism

K_{*}^{M W} (F) \to K_{*}^{M} (F)

with kernel

(η)

. We write

[u_{1}, \dots, u_{n}]

for the product

[u_{1}] \dots [u_{n}]

Theorem 4.2 (Hopkins-Morel [92, THEOREM 6.4.5]). Let

I (F) \subset G W (F)

be the kernel of the rank homomorphism

G W (F) \to Z

, with the nth power ideal

I^{n} (F) \subset G W (F)

for

n > 0

. Define

I^{n} (F) = W (F)

for

n \leq 0

. Then for each

n \in Z

, the surjection

K_{n}^{M W} (F) \to K_{n}^{M} (F)

extends to an exact sequence

0 \to I^{n + 1} (F) \to K_{n}^{M W} (F) \to K_{n}^{M} (F) \to 0

For

n = 0, K_{0}^{M} (F) = Z, K_{0}^{M W} (F)

is isomorphic to

G W (F)

and the above sequence is isomorphic to the defining sequence for

I (F)

. For

n < 0, K_{n}^{M} (F) = 0

and

K_{n}^{M W} (F) ≅

W (F)

. Finally, we have, for each

n < 0

, a commutative diagram

and, for

n = 0

, the commutative diagram

where

π

is the canonical surjection.

The isomorphism

G W (F) \tilde{\to} K_{0}^{M W} (F)

sends

⟨ u ⟩

1 + η [u]

, where

⟨ u ⟩

is the rank one form

⟨ u ⟩ (x) := u x^{2}

; since a quadratic form over

F

is diagonalizable (char

F \neq 2

), the isomorphism is completely determined by its value on the forms

⟨ u ⟩

. Given a 1-dimensional

F

-vector space

L

, we have the

GW (F)

-module

GW (F; L)

of non-degenerate,

L

-valued quadratic forms

q : V \to L

; each vector space isomorphism

ϕ : L \to F

gives an isomorphism of

GW (F)

-modules

GW (F; L) ≅ GW (F)

. Since

K_{*}^{M W} (F)

is a

Z

-graded

K_{0}^{M W} (F) = G W (F)

-module, we can form the

Z

-graded

K_{*}^{M W} (F)

-module

K_{*}^{M W} (F; L) :=

GW (F; L) \otimes_{G W (F)} K_{*}^{M W} (F)

Given a

dvr O

with residue field

k

, quotient field

F

, and generator

t

for the maximal ideal, one has the map

\partial_{t} : K_{n}^{M W} (F) \to K_{n - 1}^{M W} (k)

determined by the formulas

\partial_{t} ([t, u_{2}, \dots, u_{n}]) = [{\bar{u}}_{2}, \dots, {\bar{u}}_{n}], \partial_{t} ([u_{1}, u_{2}, \dots, u_{n}]) = 0, \partial_{t} (η \cdot x) = η \cdot \partial_{t} (x)

for

u_{1}, \dots, u_{n} \in O^{\times}

, and

x \in K_{n + 1}^{M W} (F)

, where

{\bar{u}}_{i}

is the image of

u_{i}

k^{\times}

. This is similar to the well-known boundary map

\partial : K_{n}^{M} (F) \to K_{n - 1}^{M} (k)

, with the difference, that

\partial

does not depend on the choice of

t

while

\partial_{t}

does. To get a boundary map that is independent of the choice of parameter

t

, one needs to include the twisting. This yields the well-defined boundary map

\partial : K_{n}^{M W} (F; L \otimes_{O} F) \to K_{n - 1}^{M W} (k; L \otimes_{O} {(m / m^{2})}^{\lor})

for

L

a free rank-one

O

-module, independent of the choice of generator for the maximal ideal

m

, where

\partial

is defined by choosing a generator

t

and an

O

-basis

λ

for

L

, and setting

\partial (x \otimes λ) := \partial_{t} (x) \otimes λ \otimes \partial / \partial t

Definition 4.3. Let

X

be a smooth finite type

k

-scheme, and let

L

be an invertible sheaf on

X

. The

n

L

-twisted Rost-Schmid complex for Milnor-Witt

K

-theory is the complex

{RS}^{*} (X, L, n)

with

{RS}^{m} (X, L, n) := ⨁_{x \in X^{(m)}} K_{n - m}^{M W} (k (x); L_{x} \otimes_{O_{X, x}} \overset{m}{⋀} {(m_{x} / m_{x}^{2})}^{\lor})

and boundary map

\partial^{m} : {RS}^{m} (X, L, n) \to {RS}^{m + 1} (X, L, n)

the sum of the maps

\begin{aligned} \partial_{w, x} & : K_{n - m}^{M W} (k (x); L_{x} \otimes O_{X, x} \overset{m}{⋀} {(m_{x} / m_{x}^{2})}^{\lor}) \\ \to K_{n - m - 1}^{M W} (k (w); L_{x} \otimes \otimes_{X, x} \overset{m + 1}{⋀} {(m_{w} / m_{w}^{2})}^{\lor}) \end{aligned}

associated to the normalization of the local ring

O_{\bar{x}, w}

for

w \in \bar{x} \cap X^{(m + 1)}

. Here we have cheated a bit in the definition of

\partial_{w, x}

. This is correct if

O_{\bar{x}, w}

is a dvr, which is the case outside of finitely many points

w \in \bar{x} \cap X^{(m + 1)}

; in general, one needs to use a push-forward map in Milnor-Witt

K

-theory for finite field extensions to define

\partial_{w, x}

The twisted Milnor-Witt sheaf

K_{n}^{M W} (L)_{X}

is the Nisnevich sheaf on

X

associated to the presheaf

U \mapsto H^{0} ({RS}^{*} (U, L \otimes O_{U}, n))

The codimension

n

twisted Chow-Witt group of

X, \tilde{{C H}^{n}} (X; L)

, is defined as

{\tilde{C H}}^{n} (X; L) := H^{n} ({RS}^{*} (X, L, n))

For details, see [93, CHAP. 5] or [31, cHAP. 2].

For Milnor

K

-theory, one has the Gersten complex

G^{*} (X, n)

\begin{aligned} G^{*} (X, n) := ⨁_{x \in X^{(0)}} K_{n}^{M W} (k (x)) \overset{\partial^{0}}{\to} \dots \overset{\partial^{n - m + 1}}{\to} ⨁_{x \in X^{(m)}} K_{n - m}^{M} (k (x)) \\ \overset{\partial^{n - m}}{\to} \dots \overset{\partial^{n - 1}}{\to} ⨁_{x \in X^{(n)}} K_{0}^{M} (k (x)), \end{aligned}

with essentially the same definition as the Rost-Schmid complex, without the twisting. This gives us the Milnor

K

-theory sheaf

K_{n, X}^{M} := \ker \partial^{0}

, and it follows easily from the definitions that

{C H}^{n} (X) = H^{n} (G^{*} (X, n))

. The same ideas that lead to the Bloch-Kato formula [78]

{C H}^{n} (X) ≅ H^{n} (X_{N i s}, K_{n, X}^{M})

give the isomorphism

{\tilde{C H}}^{n} (X; L) ≅ H^{n} (X_{N i s}, K_{n}^{M W} (L)_{X})

(see the discussion following [31, DEFINITION 3.1] for details). The maps

K_{n}^{M W} \to K_{n}^{M}

give the map of complexes

{RS}^{*} (X, L, n) \to G^{*} (X, n)

and the corresponding map rnk

_{X, n}

{\tilde{C H}}^{n} (X; L) \to {C H}^{n} (X)

The twists by an invertible sheaf are not just a device for defining the Rost-Schmid complexes and the Chow-Witt groups, they play an integral part in the structure of the overall theory. The Chow groups of smooth varieties admit the functorialities of a Borel-Moore homology theory: they have functorial pullback maps

f^{*} : {C H}^{n} (Y) \to {C H}^{n} (X)

for each morphism

f : X \to Y

{Sm}_{k}

, and for

f : X \to Y

a proper morphism of relative dimension

d

, one has the functorial proper push-forward map

f_{*} : {C H}^{n} (X) \to {C H}^{n - d} (Y)

. The ChowWitt groups also have a contravariant functoriality; for

f : X \to Y

, and

L

an invertible sheaf on

Y

, one has the functorial pullback

f^{*} : {\tilde{C H}}^{n} (Y, L) \to {\tilde{C H}}^{n} (X, f^{*} L)

But for the proper push-forward, one needs to include the orientation sheaf, this being the usual relative dualizing sheaf

ω_{f} := ω_{X / k} \otimes f^{*} ω_{Y / k}^{- 1}

, where

ω_{X / k} := \det Ω_{X / k}^{1}

is the sheaf of top-dimensional forms. The push-forward takes the form

f_{*} : {\tilde{C H}}^{n} (X, ω_{f} \otimes f^{*} L) \to {\tilde{C H}}^{n - d} (Y, L)

This limits the possible twists

{\tilde{C H}}^{n} (X, M)

for which a push-forward

f_{*}

is even defined; this type of restricted push-forward is typical of so-called SL-oriented theories, such as hermitian

K

-theory. See [1] for a detailed discussion of SL-oriented theories and [31, chAP. 3] for the details concerning the push-forward in

{\tilde{C H}}^{*}

4.2. The homotopy $t$ -structure and Morel's theorem

Building on the Bloch-Kato formula,

{C H}^{n} (X) ≅ H^{n} (X_{N i s}, K_{n, X}^{M})

, one can construct a good bigraded cohomology theory

EM {(K_{*}^{M})}^{* *}

by using all the cohomology groups. To get the correct bigrading, one should set

EM {(K_{*}^{M})}^{a, b} (X) := H^{a - b} (X_{N i s}, K_{b}^{M})

giving in particular

EM {(K_{*}^{M})}^{2 n, n} (X) = {C H}^{n} (X)

. It was recognized early on that this theory is not the sought-after motivic cohomology, for instance, for

X = Spec F, F

a field, one gets exactly the Milnor

K

-theory of

F

, and none of the other parts of the

K

-theory of

F

. In spite of this, this theory and the similarly defined theory for Milnor-Witt

K

-theory have a natural place in the universe of motivic cohomology theories, which we now explain.

The classical stable homotopy category

S H

is a triangulated category with a natural

t

-structure measuring connectedness, mentioned in Section 2.5. For

S H

, the truncations give the terms in the Moore-Postnikov tower

\dots \to τ_{\geq n + 1} E \to τ_{\geq n} E \to \dots \to E

with

τ_{\geq n} E \to E

characterized by inducing an isomorphism on

π_{m}

for

m \geq n

and with

π_{m} τ_{\geq n} E = 0

for

m < n

. The heart of SH is the category of spectra

E

with

π_{m} E = 0

for

m \neq 0

, which are just the Eilenberg-MacLane spectra

EM (A), A

an abelian group. Thus, the heart of

S H

A b

and the cohomology theory represented by

τ_{0} E

EM {(π_{0} E)}^{n} (X) := H^{n} (X, π_{0} E)

singular cohomology with coefficients in the abelian group

π_{0} E

We have a parallel

t

-structure on

S H (k)

, introduced by Morel [92, Â§5.2], called the homotopy

t

-structure (and not coming from Voevodsky's slice tower discussed in Section 2.5). This is similar to the

t

-structure on

S H

, where one takes into account the fact that one has bigraded homotopy sheaves

π_{a, b} E

for

E \in S H (k)

, rather than a

Z

-graded family of homotopy groups

π_{n} E

for

E \in S H

. The truncation

τ_{\geq n} E

is characterized by

π_{a, b} (τ_{\geq n} E) = {\begin{cases} π_{a, b} (E) & if a - b \geq n \\ 0 & if a - b < n \end{cases}

Recalling that the sphere

S^{a, b}

S^{a - b} \land G_{m}^{b}

, the homotopy

t

-structure on

S H (k)

is measuring

S^{1}

-connectedness, instead of the

P^{1}

-connectedness measured by Voevodsky's slice tower.

We denote the 0 th truncation

τ_{0} E

for

E \in S H (k)

EM (π_{- *, - *} E)

; the notation comes from Morel's identification of the heart with his category of homotopy modules; for details, see [92, Â§5.2]. The corresponding cohomology theory satisfies, for

X \in {S m}_{k}

EM {(π_{- *, - *} E)}^{a, b} (X) = H^{a - b} (X_{N i s}, π_{- b, - b} (E))

Here we have Morel's fundamental theorem [92, THEOREM 6.4.1] computing

τ_{0}

of the sphere spectrum

1_{k} \in S H (k)

Theorem 4.4 (Morel). Let

k

be a perfect field. Then there are canonical isomorphisms of sheaves on

{S m}_{k}

π_{- n, - n} (1_{k}) = K_{n}^{M W}

Consequently,

τ_{0} 1_{k} = EM (K_{*}^{M W})

and

EM {(K_{*}^{M W})}^{a, b} (X) = H^{a - b} (X_{N i s}, K_{b, X}^{M W})

Going back in time a bit, we have the theorem of Totaro [115] and Nesterenko-Suslin [96]

H^{n} (F, Z (n)) ≅ K_{n}^{M} (F)

for

F

a field. Combined with the isomorphism

s_{0} 1_{k} ≅ H Z

[9, 85, 122]

, we have

Theorem 4.5. Let

k

be a perfect field. Then

τ_{0} s_{0} 1_{k} = τ_{0} H Z = EM (K_{*}^{M})

and

EM {(K_{*}^{M})}^{a, b} (X) = H^{a - b} (X_{N i s}, K_{b, X}^{M})

for

X \in {Sm}_{k}

Bachmann proves an extension of this result. Recall Voevodsky's slice tower

\dots \to f_{n + 1} E \to f_{n} E \to \dots \to f_{0} E \to \dots \to E

with

s_{n} E

the layer given by the distinguished triangle

f_{n + 1} E \to f_{n} E \to s_{n} E \to f_{n + 1} E [1]

Recall that this is not the truncation tower of a

t

-structure, as the subcategories defined by the layers

s_{n} := f_{n} / f_{n + 1}

are triangulated categories, not abelian categories.

Proposition 4.6 ([7, LEMMA 12]). Let

1_{k} \to EM (K_{*}^{M})

be the composition

1_{k} \to τ_{0} 1_{k} =

EM (K_{*}^{M W}) \to EM (K_{*}^{M})

, the latter map induced by the surjection

K_{*}^{M W} \to K_{*}^{M}

. Then the induced maps

s_{0} (1_{k}) \to s_{0} EM (K_{*}^{M}) \leftarrow f_{0} EM (K_{*}^{M}) = f_{0} τ_{0} H Z

are all isomorphisms, so all of these objects are isomorphic to the motivic cohomology spectrum

H Z

The truncation functors for the homotopy

t

-structure and for the Voevodsky slice tower do not commute. Since

1_{k}

is effective, we have

f_{0} 1_{k} = 1_{k}

and so

τ_{0} f_{0} 1_{k} = τ_{0} 1_{k} =

EM (K_{*}^{M W})

. The truncations in the other order give us something new.

4.3. Milnor-Witt motivic cohomology

Definition 4.7 ([7, notation, P. 1134, JUSt BEFORE LEMMA 12]). Let

k

be a perfect field. Define the Milnor-Witt motivic cohomology spectrum

\tilde{H} Z \in S H (k)^{eff}

\tilde{H} Z := f_{0} (τ_{0} 1_{k}) = f_{0} E M (K_{*}^{M W})

The canonical map

τ_{0} 1_{k} \to τ_{0} s_{0} 1_{k} = τ_{0} H Z

induces the map

\tilde{H} Z = f_{0} (τ_{0} 1_{k}) \overset{Ξ}{\to} f_{0} τ_{0} H Z = H Z .

For

X \in {Sm}_{k}

, the Milnor-Witt motivic cohomology in bidegree

(a, b)

is defined as

\tilde{H} Z^{a, b} (X)

Remarkably, one can compute

\tilde{H} Z^{a, b} (X)

in terms of the Milnor-Witt sheaves, at least for some of the indices

(a, b)

; one also recovers the Chow-Witt groups. For

X = Spec F

, the spectrum of a field

F

, one has a complete computation in terms of the Milnor-Witt

K

-groups and the usual motivic cohomology

H Z^{a, b} (X) := H^{a} (X, Z (b))

Theorem 4.8 (Bachmann). For

X \in {Sm}_{k}

and

b \leq 0

, there are natural isomorphisms

\tilde{H} Z^{a, b} (X) ≅ H^{a - b} (X_{N i s}, K_{b, X}^{M W}) = {\begin{cases} H^{a - b} (X_{N i s}, W_{X}) & for b < 0 \\ H^{a - b} (X_{N i s}, E W_{X}) & for b = 0 \end{cases}

Here

W_{X}

is the sheaf of Witt groups and

E W_{X}

is the sheaf of Grothendieck-Witt rings.

For

X \in {S m}_{k}

and

n \in Z

, we have

\tilde{H} Z^{2 n, n} (X) ≅ {\tilde{C H}}^{n} (X)

For

F

a field, we have isomorphisms

\tilde{H} Z^{a, b} (Spec F) ≅ {\begin{cases} K_{n}^{M W} (F) & for a = b = n \\ H Z^{a, b} (Spec F) & for a \neq b \end{cases}

This follows from

Theorem 4.9 ([7, THEOREM 17]). Let

\tilde{H} Z^{a, b}, H Z^{a, b}

denote the respective homotopy sheaves

π_{- a, - b} (\tilde{H} Z), π_{- a, - b} (H Z)

. Then for

a \neq b

, the map

Ξ^{a, b} : \tilde{H} Z^{a, b} \to H Z^{a, b}

is an isomorphism. Moreover, we have canonical isomorphisms

\tilde{H} Z^{b, b} = K_{b}^{M W}

H Z^{b, b} = K_{b}^{M}

, and

Ξ^{a, b} : \tilde{H} Z^{b, b} \to H^{b, b}

is canonical surjection

K_{b}^{M W} \to K_{b}^{M}

To prove Theorem 4.8, one applies this to the local-global spectral sequence

E_{2}^{p, q} (n) := H^{p} (X_{N i s}, \tilde{H} Z^{q, n}) \Rightarrow \tilde{H} Z^{p + q, n} (X)

noting that

H Z^{q, n} = 0

for

n < 0

. This implies that the Gersten resolution of

H Z^{q, n}

has length

\leq n

and thus

H^{p} (X_{Nis}, H Z^{q, n}) = 0

for

p > n

In general, one can approximate

\tilde{H} Z^{a, b} (X)

using the local-global sequence. Combined with Theorem 4.9 and the exact sheaf sequence

0 \to ℓ^{n + 1} \to K_{n}^{M W} \to K_{n}^{M} \to 0

this tells us that the Milnor-Witt cohomology of

X

is built out of the usual motivic cohomology combined with information arising from quadratic forms.

4.4. Milnor-Witt motives

Rather than pulling the Milnor-Witt cohomology out of the motivic stable homotopy hat, there is another construction that is embedded in a Voevodsky-type triangulated category built out of a modified category of correspondences. We refer to [8] and [31] for details.

The Chow-Witt groups on a smooth

X

have been defined using the Rost-Schmid complex; one can also define Chow-Witt cycles with a fixed support using a modified version of the Rost-Schmid complex.

Definition 4.10. Let

X

be a smooth

k

-scheme,

L

an invertible sheaf on

X

, and

T \subset X

a closed subset. The

n

L

-twisted Rost-Schmid complex with supports in

T, {RS}_{T}^{*} (X, n; L)

, is the subcomplex of

{RS}^{*} (X, L, n)

with

{RS}_{T}^{m} (X, L, n) := ⨁_{x \in T \cap X^{(p)}} K_{n - m}^{M W} (k (x); L_{x} \otimes_{O_{X, x}} \overset{m}{⋀} {(m_{x} / m_{x}^{2})}^{\lor}) \subset {RS}^{m} (X, L, n)

The usual arguments used to prove Gersten's conjecture yield the following result.

Lemma 4.11. Let

X

be a smooth

k

-scheme,

L

an invertible sheaf on

X

, and

T \subset X

a closed subset. The cohomology with support

H_{T}^{p} (X, K_{n}^{M W} (L)_{X})

is computed as

H_{T}^{p} (X, K_{n}^{M W} (L)_{X}) = H^{p} ({RS}_{T}^{*} (X, L, n))

Suppose

T

has pure codimension

n

X

. Then

X^{(m)} \cap T = \emptyset

for

m < n, X^{(n)} \cap T

is the finite set of generic points

T^{(0)}

T

and

X^{(n + 1)} \cap T = T^{(1)}

is the set of codimension one points of

T

. This gives us the exact sequence

\begin{aligned} 0 & \to H_{T}^{n} (X, K_{n}^{M W} (L)_{X}) \to ⨁_{z \in T^{(0)}} G W (k (z), \det^{- 1} (m_{z} / m_{z}^{2}) \otimes L) \\ \to ⨁_{z \in T^{(1)}} W (k (z), \det^{- 1} (m_{z} / m_{z}^{2}) \otimes L) \end{aligned}

which allows us to think of

H_{T}^{n} (X, K_{n}^{M W} (L)_{X})

as the group of "Grothendieck-Witt cycles" supported on

T

, whose definition we hinted at in the beginning of this section. We write this as

{\tilde{Z}}_{T}^{n} (X, L, n)

, with the warning that this is only defined for

T

a closed subset of a smooth

X

of pure codimension

n

Note that the fact that

T

has pure codimension

n

implies that there are no relations in

H_{T}^{n} (X, K_{n}^{M W} (L)_{X})

coming from

K_{1}^{M W} (k (w))

for

w

a codimension

n - 1

point of

X

. For similar reasons, the corresponding group for the Chow groups,

H_{T}^{n} (X, K_{n, X}^{M})

, is just the subgroup

Z_{T}^{n} (X)

Z^{n} (X)

freely generated by the irreducible components of

T

, that is, the group of codimension

n

cycles on

X

with support contained in

T

For

T \subset T^{'} \subset X

, two codimension-

n

closed subsets, we have the evident map

{\tilde{Z}}_{T}^{n} (X, L, n) \to {\tilde{Z}}_{T^{'}}^{n} (X, L, n)

. The rank map

G W (-) \to Z

gives the homomorphism

{\tilde{Z}}_{T}^{n} (X, L, n) \to Z_{T}^{n} (X)

Definition 4.12. For

X, Y

{Sm}_{k}

, let

A (X, Y)

be the set of closed subsets

T \subset X \times Y

such that each component of

T

is finite over

X

and maps surjectively onto an irreducible component of

X

. We make

A (X, Y)

a poset by the inclusion of closed subsets.

Note that if

Y

is irreducible of dimension

n

, then a closed subset

T \subset X \times Y

is in

A (X, Y)

if and only if

T

is finite over

X

and has pure codimension

n

X \times Y

Definition 4.13 (CalmÃ¨s-Fasel

[31, § 4.1]

). Let

X, Y

be in

{Sm}_{k}

and suppose

Y

is irreducible of dimension

n

. Define

{\tilde{Corr}}_{k} (X, Y) = {colim}_{T \in A (X, Y)} {\tilde{Z}}_{T}^{n} (X \times Y, p_{2}^{*} ω_{Y / k})

Extend the definition to general

Y

by additivity.

Using the functorial properties of pullback, intersection product and proper pushforward for the Chow-Witt groups with support, we have a well-defined composition law

{\tilde{Corr}}_{k} (Y, Z) \times {\tilde{Corr}}_{k} (X, Y) \to {\tilde{Corr}}_{k} (X, Z)

via the same formula used to define the composition in

{C o r}_{k}

Z_{2} \circ Z_{1} := p_{X Z *} (p_{Y Z}^{*} (Z_{2}) \cap p_{X Y}^{*} (Z_{1}))

The twisting by the relative dualizing sheaf in the definition of

{\tilde{Corr}}_{k} (-, -)

is exactly what is needed for the push-forward map

p_{X Z} *

to be defined.

This defines the additive category

{\tilde{Corr}}_{k}

with objects

{S m}_{k}

and morphisms

{\tilde{Corr}}_{k} (X, Y)

. The rank map gives an additive functor

rnk : {\tilde{Corr}}_{k} \to {Cor}_{k}

One then follows the program used by Voevodsky to define the abelian category of Nisnevich sheaves with Milnor-Witt transfers,

{Sh}_{Nis}^{M W} (k)

, and then

{\tilde{D M}}^{eff} (k) \subset

D ({Sh}_{N i s}^{M W t r} (k))

as the full subcategory of complexes with strictly

A^{1}

-homotopy invariant cohomology sheaves. One has the localization functor

{\tilde{L}}_{A^{1}} : D ({S h}_{N i s}^{M W t r} (k)) \to {\tilde{D M}}^{e f f} (k)

constructed using the Suslin complex, the representable sheaves

{\tilde{Z}}^{t r} (X)

for

X \in {Sm}_{k}

, their corresponding motives

{\tilde{M}}^{eff} (X) := {\tilde{L}}_{A^{1}} ({\tilde{Z}}^{tr} (X)) \in {\tilde{D M}}^{eff} (k)

and the Tate motives

\tilde{Z} (n)

arising from the reduced motive of

P^{1}

. Finally, one constructs

\tilde{D M} (k)

as a category of

\tilde{Z} (1)

{\tilde{M}}^{eff} (X)

Definition 4.14. For

X \in {S m}_{k}

, categorical Milnor-Witt cohomology is

H^{p} (X, \tilde{Z} (q)) := {Hom}_{\tilde{D M} (k)} (\tilde{M} (X), \tilde{Z} (q) [p])

Theorem 4.15. There is a natural isomorphism

H^{p} (X, \tilde{Z} (q)) ≅ H {\tilde{Z}}^{p, q} (X)

The proof is very much the same as for motivic cohomology. One shows there is an equivalence of

\tilde{D M} (k)

with the homotopy category of

H \tilde{Z}

-modules (this is [8, THEOREM 5.2]). This gives an adjunction

H \tilde{Z} \land - : S H (k) ⇄ \tilde{D M} (k) : \tilde{E M}

with

H \tilde{Z} \land -

the free

H \tilde{Z}

module functor and the Eilenberg-MacLane functor

\tilde{E M}

the forgetful functor. This gives

\tilde{EM} (\tilde{Z} (0)) = H \tilde{Z}, \tilde{M} (X) = H \tilde{Z} \land Σ_{P^{1}}^{\infty} X_{+}

, and induces the isomorphism

\begin{aligned} H^{p} (X, \tilde{Z} (q)) & = {Hom}_{\tilde{D M} (k)} (\tilde{M} (X), \tilde{Z} (q) [p]) \\ ≅ {Hom}_{S H (k)} (Σ_{P^{1}}^{\infty} X_{+}, Σ^{p, q} H \tilde{Z}) = H {\tilde{Z}}^{p, q} (X) \end{aligned}

5. CHOW GROUPS AND MOTIVIC COHOMOLOGY WITH MODULUS

Up to now, all the version of motivic cohomology we have considered share the

A^{1}

homotopy invariance property, namely, that

H^{*} (X, Z (*)) ≅ H^{*} (X \times A^{1}, Z (*))

; essentially by construction, this property is enjoyed by all theories that are represented in the motivic stable homotopy category. Although this is a fundamental property controlling a large collection of cohomology theories, this places a serious restriction in at least two naturally occurring areas.

One is the use of deformation theory. This relies on having useful invariants defined on non-reduced schemes, but a cohomology theory that satisfies

A^{1}

-invariance will not distinguish between a scheme and its reduced closed subscheme. The second occurs in ramification theory. An

A^{1}

-homotopy invariant theory will not detect Artin-Schreyer covers, and would not give invariants that detect wild ramification.

Fortunately, we have an interesting cohomology theory that is not

A^{1}

-homotopy invariant, namely, algebraic

K

-theory, that we can use as a model for a general theory. Algebraic

K

-theory does satisfy the

A^{1}

-invariance property when restricted to regular schemes, but in general this fails. Besides allowing

K

-theory to have a role in deformation theory and ramification theory, this lack of

A^{1}

-invariance gives rise to interesting invariants of singularities.

5.1. Higher Chow groups with modulus

The theory of Chow groups with modulus attempts to refine the classical theory of the Chow groups to be useful in both of these areas. This is still a theory in the process of development; just as in the early days of motivic cohomology, many approaches are inspired by properties of algebraic

K

-theory.

The tangent space to the functor

X \mapsto O_{X}^{\times}

is given by the structure sheaf,

X \mapsto O_{X}

, via the isomorphism

O_{X [ε] / (ε^{2})}^{\times} ≅ O_{X}^{\times} \oplus ε \cdot O_{X}

Via the isomorphism

Pic (X) ≅ H^{1} (X, O_{X}^{\times})

, this shows that the tangent space at

X

to the functor

Pic (-)

H^{1} (X, O_{X})

In [23], Bloch computes the tangent space to

K_{2}

(on local

Q

-algebras), giving the isomorphism of sheaves on

X_{Z a r}

(for

X

Q

-scheme)

K_{2, X [ε] / (ε^{2})} ≅ K_{2, X} \oplus Ω_{X}

where

Ω_{X}

is the sheaf of absolute KÃ¤hler differentials. Bloch then uses his formula from [22],

H^{2} (X_{Z a r}, K_{2}) ≅ {C H}^{2} (X)

to justify defining

{C H}^{2} (X [ε] / (ε^{2}))

H^{2} (X [ε] / {(ε^{2})}_{Z a r}, K_{2})

, giving

{C H}^{2} (X [ε] / (ε^{2})) = {C H}^{2} (X) \oplus H^{2} (X, Ω_{X})

For

X

a smooth projective surface over

C

with

H^{2} (X, O_{X}) \neq 0

, the exact sheaf sequence

0 \to Ω_{C / Q} \otimes_{C} O_{X} \to Ω_{X} \to Ω_{X / C} \to 0

along the fact that

Ω_{C / Q}

is a

C

-vector space of uncountable dimension show that

Ω_{C / Q} \otimes_{C} H^{2} (X, O_{X})

makes a huge contribution to the tangent space

H^{2} (X, Ω_{X})

{C H}^{2} (-)

X

. This is reflected in Mumford's result [95], that if

H^{2} (X, O_{X}) ≅ H^{0} (X, Ω_{X / C}^{2})

is nonzero, then

{C H}^{2} (X)

is "infinite-dimensional," and gives some evidence for Bloch's conjecture [23, CONJECTURE (0.4)] on 0 -cycles on surfaces

X

with

H^{0} (X, Ω_{X / C}^{2}) = 0

The algebraic cycles have disappeared in this approach to Chow groups of nonreduced schemes. Bloch and Esnault [26] gave the first construction of a cycle-theoretic theory that could say something interesting about higher cycles on the non-reduced scheme Spec

k [ε] / (ε^{2})

. In a second paper [27], they modified and extended this construction to give a theory of additive higher Chow groups with modulus

m

, for the field

k

. This was motivated by Bloch's earlier use of

K

-theory on the affine line, relative to

{0, 1}

, to study

K_{3}

. Letting 1 tend to 0 , they were led to consider the relative

K

-theory space

K (k [ε], (ε^{2}))

, this being the homotopy fiber of the restriction map

K (k [ε]) \to K (k [ε] / ε^{2})

, whose homotopy groups are the relative

K

-theory groups

K_{n} (k [ε], (ε^{2}))

. Replacing 2 with

m \geq 2

gives the relative

K

-theory groups

K_{n} (k [ε], (ε^{m}))

. This led to the consideration of a complex of cycles on Spec

k [ε]

, with an additional condition imposed on the

m

th order limiting behavior of the cycles; an explicit construction of such a cycle complex with modulus,

z^{q} (k, *, m)

was given in [27]. The homology

A {C H}^{q} (k, p, m) := H_{p} (z^{q} (k, *, m))

defines the additive codimension

q

higher Chow groups with modulus

m

for Spec

k

. Bloch-Esnault recover the computation

A {C H}^{n} (k, n - 1, 2) ≅ Ω_{k}^{n - 1}

from [26], and relate the additive analogue of weight two

K_{3}

A {C H}^{2} (k, 2, 2)

, with the additive dilogarithm of Cathelineau [32].

RÃ¼lling [104] studied the projective system

\dots \to A {C H}^{n} (k, n - 1, m + 1) \to A {C H}^{n} (k, n - 1, m) \to \dots

He showed this is endowed with additional endomorphisms

F_{n}

and

V_{n}

, and the graded group

⨁_{n} A {C H}^{n} (k, n - 1, * + 1)_{* \geq 2}

has the structure of a pro-differential graded algebra. In fact, we have

Theorem 5.1 (RÃ¼lling). Let

k

be a field of characteristic

\neq 2

. The pro-dga

⨁_{n} A {C H}^{n} (k

n - 1, * + 1)

, with

F_{n}

as Frobenius and

V_{n}

as Verschiebung, is isomorphic to the de RhamWitt complex of Madsen-Hesselholt,

⨁_{n} A {C H}^{n} (k, n - 1, * + 1) ≅ ⨁_{n} W_{*} Ω_{k}^{n - 1}

With essentially the same definition as given by Bloch-Esnault, the additive cycle complex and additive Chow groups were extended to arbitrary

k

-schemes

Y

by Park [98], replacing

A^{1}

and divisor

m \cdot 0

with the scheme

Y \times A^{1}

and divisor

m \cdot Y \times 0

. Binda and Saito [20] went one step further, defining complexes

z^{q} (X, D, *)

for a pair

(X, D)

of a finite type separated

k

-scheme

X

and a Cartier divisor

D

, using essentially the same definition as before. The homology is the higher Chow group with modulus

{C H}^{q} (X, D, p) := H_{p} (z^{q} (X, D, *))

The constructions of Bloch-Esnault, Park, and Binda-Saito all use a cubical model of Bloch's cycle complex. Here one replaces the algebraic

n

-simplex,

Δ_{k}^{n} = Spec k [t_{0}, \dots

t_{n}] / \sum_{i} t_{i} - 1

, with the algebraic

n

-cube

◻^{n} := {(P^{1} ∖ {1}, 0, \infty)}^{n}

The notation means that one considers

{(P^{1} ∖ {1})}^{n} ≅ A^{n}

with its "faces" defined by setting some of the coordinates equal to 0 or

\infty

. The corresponding cycle complex

z^{q} (X, *)_{c}

has degree

n

component

z^{q} (X, n)_{c}

the codimension

q

cycles on

X \times ◻^{n}

that intersect

X \times F

properly for all faces

F

◻^{n}

; one also needs to quotient out by the degenerate cycles, these being the ones that come by pullback via projection to a

◻^{m}

with

m < n

. The differential is again an alternating sum of restrictions to the maximal faces

t_{i} = 0

and

t_{i} = \infty

This complex also computes the motivic cohomology of

X

, just as Bloch's simplicial cycle complex does. In the Binda-Saito construction, the modulus condition arises by considering the closed box

{\bar{◻}}^{n} := {(P^{1})}^{n}

. Let

F_{n}^{i} \subset {(P^{1})}^{n}

be the divisor defined by

t_{i} = 1

and let

F_{n} = \sum_{i = 1}^{n} F_{n}^{i}

. In

{(P^{1})}^{n} \times X

we have two distinguished Cartier divisors,

{(P^{1})}^{n} \times D

and

F_{n} \times X

. A subvariety

Z \subset {(P^{1} ∖ {1})}^{n} \times X

that is in

z^{q} (X, n)_{c}

satisfies the modulus condition if

p^{*} (F_{n} \times X) \geq p^{*} ({(P^{1})}^{n} \times D)

where

p : {\bar{Z}}^{N} \to {(P^{1})}^{n} \times X

is the normalization of the closure of

Z

{\bar{◻}}^{n} \times X

. Restricting to the subgroup of

Z^{q} (◻^{n} \times X)

generated by codimension

q

subvarieties

Z \subset ◻^{n} \times X

that intersect faces properly and satisfy the modulus condition yields the cycle complex with modulus

z^{q} (X; D, *) \subset z^{q} (X, *)_{c}

; the higher Chow groups with modulus is then defined as

{C H}^{q} (X; D, p) := H_{p} (z^{q} (X; D, *))

The second construction of Bloch-Esnault, and Park's generalization, are recovered as the special cases

X = A_{k}^{1}

and

D = m \cdot 0

in the Bloch-Esnault version and

X = Y \times A^{1}

D = m \cdot Y \times 0

in Park's version.

For

X

a finite type

k

-scheme, recall the Bloch motivic complex

Z_{B I} (q)_{X}^{*}

defined as the Zariski sheafification of the presheaf

U \mapsto z^{q} (X, 2 q - *)

(this is already a Nisnevich sheaf). Bloch's cycle complexes satisfy an important localization property: the natural maps to Zariski and Nisnevich hypercohomology

H^{p} (z^{q} (X, 2 q - *)) \to H^{p} (X_{Z a r}, Z_{B l} (q)_{X}^{*}) \to H^{p} (X_{N i s}, Z_{B l} (q)_{X}^{*})

are isomorphisms. This fails for the cycle complex with modulus, although the comparison between the Zariski and Nisnevich hypercohomology seems to be still an open question.

Iwasa and Kai consider the Nisnevich sheafification

Z (q)_{(X; D)}^{*}

of the presheaf

U \mapsto z^{q} (U; D \times_{X} U, 2 q - *)

We call

H^{p} (X_{N i s}, Z (q)_{(X; D)}^{*})

the motivic cohomology with modulus for

(X, D)

. Kai [74] shows that this sheafified version has contravariant functoriality. Iwasa and Kai [67] construct Chern class maps from relative

K

-theory

c_{p, q} : K_{2 q - p} (X; D) \to H^{p} (X_{N i s}, Z (q)_{X, N i s}^{*})

5.2. 0 -cycles with modulus and class field theory

There is a classical theory of 0 -cycles on a smooth complete curve

C

with a modulus condition at a finite set of points

S

, due to Rosenlicht and Serre [109, III]. The idea is quite simple, instead of relations coming from divisors (zeros minus poles) of an arbitrary rational function

f, f

is required to have a power series expansion at each point

p \in S

, with leading term 1 and the next nonzero term of the form

u t_{p}^{n_{p}}

, with

u (p) \neq 0, t_{p}

a local coordinate at

p

and the integer

n_{p} > 0

being the "modulus." This is applied to the class field theory of a smooth open curve

U \subset C

over a finite field [109, THEOREM 4], that identifies the inverse limit of the groups of degree 0 cycle classes on

U

, with modulus supported in

C ∖ U

, with the kernel of the map

π_{1}^{et} (U)^{a b} \to Gal (\bar{k} / k)

In their class field theory for higher-dimensional varieties, Kato and Saito [80] introduce a group of 0 -cycles on a

k

-scheme

X

with modulus

D

, defined by

{C H}_{0} (X, D) := H^{n} (X, K_{n, (X, D)}^{M})

with

K_{n, (X, D)}^{M}

a relative version of the Milnor

K

-theory sheaf, recalling Kato's isomorphism

H^{n} (X, K_{n}^{M}) ≅ {C H}^{n} (X)

for

X

a smooth

k

-scheme [78]. Kerz and Saito give a different definition of a group of relative 0 -cycles

C (X, D)

on a normal

k

-scheme

X

with effective Cartier
divisor

D

such that

X ∖ D

is smooth. It follows from their comments in [81, DEFINITION 1.6] that

C (X, D) = {C H}^{n} (X; D, 0)

for

X

of dimension

n

, and it is easy to see that the KatoSaito and Kerz-Saito relative 0 -cycles agree with the Rosenlicht-Serre groups in the case of curves.

Kerz and Saito consider a smooth finite-type

k

-scheme

U

, choose a normal compactification

X

and define the topological group

C (U) := lim_{D} C (X, D)

, as

D

runs over effective Cartier divisors on

X

, supported in

X ∖ U

, and with each

C (X, D)

given the discrete topology. They show that

C (U)

is independent of the choice of

X

, and their main result generalizes class field theory for smooth curves over a finite field as described above.

Theorem 5.2 ([81, Ñ‚HEORem 3.3]). Let

k

be a finite field of characteristic

\neq 2

and let

U

be a smooth variety over

k

, Then

C (U)

is isomorphic as topological group to a dense subgroup of the abelianized Ã©tale fundamental group

π_{1}^{e t} (U)^{a b}

and this isomorphism induces an isomorphism of the degree 0 part

C (U)^{0}

C (U)

with the kernel

π_{1}^{e t} (U)_{0}^{a b}

π_{1}^{e t} (U)^{a b} \to π_{1}^{e t} (k)

5.3. Categories of motives with modulus

There has been a great deal of interest in constructing a categorical framework for motivic cohomology with modulus. A central issue is the lack of

A^{1}

-homotopy invariance for this theory, which raises the question of what type of homotopy invariance should replace this.

One direction has been the construction of a reasonable replacement for the category of homotopy invariant Nisnevich sheaves with transfers. A non-homotopy invariant version has been developed via the theory of reciprocity sheaves, the name coming from the reciprocity laws in class field theory of curves and its relation to the group of 0 -cycles with modulus of Rosenlicht-Serre. We will say a bit about reciprocity sheaves later on, in the context of motives for log schemes Section 5.4.

For now, we will look at categories of motives with modulus constructed on the Voevodsky model by introducing a new notion of correspondence and a suitable replacement for

A^{1}

-homotopy invariance.

Looking at algebraic

K

-theory, the closest replacement for

A^{1}

-homotopy invariance seems to be the

P^{1}

-bundle formula

K_{n} (X \times P^{1}) ≅ K (X) \cdot [O_{X \times P^{1}}] \oplus K (X) \cdot [O_{X \times P^{1}} (- 1)]

valid for a general scheme

X

. This has led to attempts to create a category of motives with

Here one has the problem that

P^{1}

does not have the structure of an interval, a structure enjoyed by

A^{1}

. One considers

A^{1}

together with "endpoints" 0,1 . Following the general theory of a site with interval, as developed by Morel-Voevodsky [94, chap. 2], one needs the multiplication map

m : A^{1} \times A^{1} \to A^{1}

to allow one to consider

(A^{1}, 0, 1)

as an abstract interval. In the construction of the cycle complex with modulus, one identifies

(A^{1}, 0, 1)

with

(P^{1} ∖ {1}, 0, \infty)

, and the corresponding multiplication map

m^{'} : (P^{1} ∖ {1}) \times (P^{1} ∖ {1}) \to

P^{1} ∖ {1}

only extends as a rational map

P^{1} \times P^{1} \to P^{1}

. However,

m^{'}

becomes a morphism
after blowing up the point

(1, 1)

, which suggests that one should consider the closure of the graph of

m^{'}

P^{1} \times P^{1} \times P^{1}

as an allowable correspondence from

P^{1} \times P^{1}

P^{1}

With this as starting point, Kahn, Miyazaki, Saito, and Yamazaki [69-71] follow Voevodsky's program, defining a category of modulus correspondences MCor

_{k}

. Objects are pairs (

\bar{M}, M^{\infty}

) with

\bar{M}

a separated finite-type

k

-scheme and

M^{\infty}

an effective Cartier divisor on

\bar{M}

such that the open complement

M^{\circ} := \bar{M} ∖ M^{\infty}

is smooth. The morphism group

Z

(finite and surjective over a component of

M^{\circ}

) such that

(i) The closure

\bar{Z}

Z

\bar{M} \times \bar{N}

is proper over

\bar{M}

(not necessarily finite).

(ii) Let

f : {\bar{Z}}^{N} \to \bar{M} \times \bar{N}

be the normalization of

\bar{Z}

. Then

f^{*} p_{1}^{*} M^{\infty} \geq f^{*} p_{2}^{*} N^{\infty}

The composition law in

{C o r}_{k}

preserves conditions (i) and (ii), giving the category

{\underline{M C o r}}_{k}

with functor

\underline{M} {Cor}_{k} \to {Cor}_{k}

sending

(\bar{M}, M^{\infty})

M^{\circ}

and with

\underline{M} {Cor}_{k} ((\bar{M}, M^{\infty})

(\bar{N}, N^{\infty})) \to {Cor}_{k} (M^{\circ}, N^{\circ})

the inclusion. The product of pairs makes

\underline{{M C o r}_{k}}

a symmetric monoidal category and the functor to

{C o r}_{k}

is symmetric monoidal.

Let

\bar{◻}

be the object

(P^{1}, {1})

. As hinted above, the closure of the graph of

m^{'} : (P^{1} ∖ {1}) \times (P^{1} ∖ {1}) \to P^{1} ∖ {1}

defines a morphism

m : \bar{◻} \times \bar{◻} \to \bar{◻}

in MCor

_{k}

They then consider the abelian category of additive presheaves of abelian groups

pairs

(X, D)

, with

X

a proper

k

-scheme, as a full subcategory of

\underline{MCor}_{k}

, with its presheaf category

{MPST}_{k}

They define a category of effective proper motives with modulus,

{MDM}^{e f f} (k)

, by localizing the derived category

D ({MPST}_{k})

. Roughly speaking, they follow the Voevodsky program, replacing the

A^{1}

-localization with

\bar{◻}

localization. To get the proper Nisnevich localization is a bit technical; we refer the reader to [71, DEFINITION 1.3.9] for details.

There is still quite a bit that is not known. One central problem is how to realize the various constructions of the higher Chow groups with modulus as morphisms in a suitable triangulated category. There is a connection, at least for the modulus version of Suslin homology and the Suslin complex, which we now describe.

One can show that the cubical version of the Suslin complex

C_{*}^{Sus} (X)_{c} (Y) := {Hom}_{{Cor}_{k}} (Y \times ◻^{*}, X) / degn

is naturally quasi-isomorphic to the simplicial version

C_{*}^{Sus} (X) (Y)

, where / degn means taking the quotient by the image of the pullback maps via the projections

Y \times ◻^{n} \to Y \times

◻^{n - 1}

. For a modulus pair

(X, D)

, one can similarly form the naive Suslin complex

C_{*}^{Sus} (X, D) (Y, E) := {Hom}_{{\underline{M C o r}}_{k}} ((Y, E) \otimes {\bar{◻}}^{*}, X) / degn

Taking

(Y, E) = (Spec k, \emptyset)

, we have the complex

C_{*}^{S u s} (X, D) := C_{*}^{S u s} (X, D) (Spec k, \emptyset)

Next, there is a derived Suslin complex

R C_{*}^{Sus} (X, D)_{c} (-)

with a natural map of presheaves

C_{*}^{S u s} (X, D)_{c} (-) \to R C_{*}^{S u s} (X, D)_{c} (-)

By [71, THEOREM 2], for

(X, D)

a proper modulus pair,

R C_{*}^{Sus} (X, D)_{c} (-)

computes the maps in

{MDM}^{e f f} (k)

H_{n} (R C_{*}^{Sus} (X, D)_{c} (Spec k, \emptyset)) = {Hom}_{{M D M}^{eff} (k)} ((Spec k, \emptyset), M^{eff} (X, D))

However, one should not expect that the Suslin complex or its derived version should yield a version of the higher Chow groups. If one looks back at the setting of

D M (k)

, the object that most naturally yields the higher Chow groups for an arbitrary finite type

k

-scheme

X

is the motive with compact supports

M^{c} (X)

. This is defined as

C_{*}^{Sus} (Z_{t r}^{c} (X))

, where

Z_{t r}^{c} (X)

is the presheaf with transfers with

Z_{t r}^{c} (X) (Y)

the free abelian group on integral

W \subset Y \times X

, with

W \to Y

quasi-finite and dominant over a component of

Y \in {Sm}_{k}

. See [127, cHAP. 5, PROPOSITION 4.2.9] for the relation of

M^{c} (X)

with Bloch's higher Chow groups.

One can define a similar version with modulus as the object

M^{c} (X, D)

associated to the presheaf

Z_{t r}^{c} (X, D)

, with

Z_{t r}^{c} (X, D) (Y, E) \subset Z_{\dim Y} (Y \times X)

the subgroup generated by closed subvarieties

W \subset (Y ∖ E) \times (X ∖ D)

that are quasi-finite and dominant over

Y

, and with the usual modulus condition, that the normalization

ν : {\bar{W}}^{N} \to

Y \times X

of the closure of

W

Y \times X

satisfies

v^{*} (E \times X) \geq v^{*} (Y \times D)

There is an analog of Suslin's comparison theorem in the affine case, due to KaiMiyazaki [75]: They define an equi-dimensional cycle complex with modulus

z_{d}^{equi} (X, D, *) \subset z_{d} (X, D, *)

which for

d = 0

is the Suslin complex with modulus

C_{*}^{Sus} (Z_{t r}^{c} (X, D)) (Spec k, \emptyset)

Theorem 5.3 (Kai-Miyazaki). Let

(X, D)

be a modulus pair, with

X

affine. Then there is a pro-isomorphism

{H_{*} (z_{d}^{equi} (X, m D, *))}_{m} ≅ {{C H}_{d} (X, m D, *)}_{m}

Miyazaki [91] has defined objects

z^{equi} (X, D, d) \in {MNST}_{k}

, with

Z_{tr}^{c} (X, D) =

z^{equi} (X, D, 0)

. The sheaf

z^{equi} (X, D, r)

is defined similarly to

Z_{t r}^{c} (X, D)

, with

z^{equi} (X, D

d) (Y, E)

the group of cycles on

(Y ∖ E) \times (X ∖ D)

generated by closed, integral

W \subset (Y ∖ E) \times (X ∖ D)

that are equi-dimensional of dimension

d

over

Y ∖ E

, dominate a component of

Y ∖ E

, and with

v : {\bar{W}}^{N} \to Y \times X

satisfying the modulus condition

v^{*} (E \times X) \geq v^{*} (Y \times D)

Moreover, for an arbitrary modulus pair

(X, D)

, one has

z_{d}^{equi} (X, D, *) = C_{*}^{Sus} (z^{equi} (X, D, d)) (Spec k, \emptyset)

and there is the canonical map

C_{*}^{Sus} (z^{e q u i} (X, D, d)) \to R C_{*}^{Sus} (z^{equi} (X, D, d))

Letting

{C H}_{q}^{equi} (X, D, p) = H_{p} (z_{q}^{equi} (X, D, *))

, we have the natural map

{C H}_{q}^{equi} (X, D, p) \to {C H}_{q} (X, D, p)

which is an isomorphism for

X

affine, and we have the natural maps for

(X, D)

a proper modulus pair

\begin{aligned} {C H}_{q}^{equi} (X, D, p) & \to H_{p} (R C_{*}^{Sus} (z^{equi} (X, D, q)) (Spec k, \emptyset)) \\ \to {Hom}_{{MDM}^{eff} (k)} (M (Spec k, \emptyset) [p], R C_{*}^{Sus} (z^{equi} (X, D, q))) \end{aligned}

For a proper modulus pair, let

M^{c} (X, D)

denote the image of

Z_{tr}^{c} (X, D)

{MDM}^{e f f} (k)

. One can ask if there are analogs of the theorem of Kahn-Miyazaki-Saito-Yamazaki.

Question 5.4. For

(X, D)

a proper modulus pair, are the maps

H_{p} (R C_{*}^{S u s} (Z_{t r}^{c} (X, D)) (Spec k, \emptyset)) \to {Hom}_{{M D M}^{e f f} (k)} (M (Spec k, \emptyset) [p], M^{c} (X, D))

isomorphisms? More generally, are the maps

\begin{aligned} H_{p} (R C_{*}^{Sus} (z^{equi} (X, D, q)) (Spec k, \emptyset)) \\ \to {Hom}_{{MDM}^{eff} (k)} (M (Spec k, \emptyset) [p], R C_{*}^{Sus} (z^{equi} (X, D, q))) \end{aligned}

isomorphisms?

It is also not clear if the map

{C H}_{q}^{equi} (X, D, p) \to H_{p} (R C_{*}^{Sus} (z^{equi} (X, D, q)) (Spec k, \emptyset))

should be an isomorphism. Possibly one should also consider the Nisnevich hypercohomology

H^{- p} (X_{N i s}, Z_{q}^{equi} (X, D, *))

, with

Z_{q}^{equi} (X, D, *)

defined by sheafifying

U \mapsto

z_{q}^{equi} (U, U \cap D, *)

For Voevodsky motives, and for

X

a finite type

k

-scheme, the motivic Borel-Moore homology is defined by

\begin{aligned} H_{p}^{B.M.} (X, Z (q)) & := {Hom}_{{D M}^{eff} (k)} (Z (q) [p], M (X)^{c}) \\ ≅ H_{p - 2 q} (z_{q}^{equi} (X, *)) ≅ H_{p - 2 q} (z_{q} (X, *)) = {C H}_{q} (X, p - 2 q) \end{aligned}

This uses the duality

M (X)^{c} ≅ M (X)^{\lor} (d) [2 d]

for

X

of dimension

d

(valid in characteristic zero, or after inverting

p

in characteristic

p > 0

), and the extension of Suslin's quasi-isomorphism

z_{q}^{equi} (X, *) ↪ z_{q} (X, *)

to arbitrary

X

. Moreover, we have

M (X)^{c} =

M (X)

for

X

smooth and proper.

However, a corresponding motivic cohomology of modulus pairs seems to need a larger category. This is hinted at by the use of the duality (in

DM (k)) M (X)^{c} ≅

M (X)^{\lor} (d) [2 d]

in the computations described above. This says in particular that each motive

M (X)

admits a "twisted" dual in

{D M}^{e f f} (k)

, more precisely, the usual evaluation
and coevaluation maps associated with a dual exist, but as maps with target or source some

Z (d) [2 d]

rather than the unit. For a general proper modulus pair

(X, D)

, this does not seem to be the case; one seems to need modulus pairs with an anti-effective Cartier divisor. Another way to say the same thing, if one looks for a proper modulus pair

(X, D^{'})

such that

{Hom}_{{MDM}^{eff} (k)} (M (X, D^{'}), Z (q) [p])

looks at all like

{C H}^{q} (X, D, 2 q - p)

for some given proper modulus pair

(X, D)

, the defining inequalities in Corr

_{k}

suggest that

D^{'}

could be

- D

. See the section "Perspectives" in [71, InTRoduction] for further details in this direction.

5.4. Logarithmic motives and reciprocity sheaves

Grothendieck motives for log schemes have been constructed in [66], where a version for mixed motives has also been constructed using systems of realizations. There the emphasis is on versions of motives for homological or numerical equivalence in the setting of

\log

schemes. In this section we discuss a recent construction of a triangulated category of log motives, by Binda-Park-Ã˜stvÃ¦r [19], that follows the Voevodsky program. We refer the reader to the lectures notes of Ogus [97] for the facts about log schemes.

Recall that a

\log

scheme is a pair

(X, α : M \to (O_{X}, \times))

consisting of a scheme

X

and a homomorphism of sheaves of commutative monoids

α : M \to (O_{X}, \times)

such that

α^{- 1} (O_{X}^{\times}) \to O_{X}^{\times}

is an isomorphism; without this last condition, the pair

(X, α : M \to

(O_{X}, \times))

is called a pre-log structure. A pre-log structure

α : M \to (O_{X}, \times)

induces a log structure

α^{\log} : M^{\log} \to (O_{X}, \times)

by taking

M^{\log}

to be the push-out (in the category of sheaves of monoids) in

Given a modulus pair

(X, D)

, there are a number of (in general distinct) induced log structures on

X

. For example, one can take the compactifying log structure, with

M := O_{X} \cap

j_{*} O_{U}^{\times}

, where

U = X ∖ D

and

j : U \to X

is the inclusion. There are other log structures, which in general depend on a choice of decomposition of

D

as a sum of effective Cartier divisors (for example, the Deligne-Faltings log structure, discussed in [97, III, DEFINITION 1.7.1]).

Replacing the category of smooth

k

-schemes is the category

1 {S m}_{k}

of fine, saturated, log smooth and separated log schemes over the log scheme Spec

k

endowed with the trivial

\log

structure. We refer the reader to [19] for details; one needs these technical conditions to construct the category of finite log correspondences. We call a separated, fine, saturated log scheme an fs log scheme.

We sketch the construction of the category of finite log correspondences, and describe how Binda-Park- stvÃ¦r follow Voevodsky's program to define the triangulated category

\log {D M}^{eff} (k)

of effective log motives over

k

For

X \in {Smm}_{k}

, let

\underline{X}

denote the underlying

k

-scheme. We let

X^{\circ} \subset \underline{X}

denote the maximal open subscheme over which the log structure

M_{X} \to O_{X}

is trivial, that is,

M_{X ∣ U} = O_{U}^{\times}

, and let

\partial X = \underline{X} ∖ X^{\circ}

Definition 5.5. 1. For

X, Y \in {Sm}_{k}

, the group

{lCor}_{k} (X, Y)

consisting of finite log correspondences from

X

Y

is the free abelian group on integral closed subschemes

\underline{Z} \subset \underline{X} \times \underline{Y}

such that

(i)

\underline{Z} \to \underline{X}

is finite and is surjective to a component of

\underline{X}

(ii) Let

Z^{N}

be the log scheme with underlying scheme the normalization

ν : {\underline{Z}}^{N} \to

X \times Y

\underline{Z}

and

\log

structure

{(ν \circ p_{1})}_{\log}^{*} M_{X} \to O_{Z^{N}}

. Here

M_{X} \to O_{X}

is the given log structure on

X

and

{(ν \circ p_{1})}_{\log}^{*} M_{X} \to O_{Z}

is the log structure induced by the pre-log structure

{(ν \circ p_{1})}^{- 1} M_{X} \to {(ν \circ p_{1})}^{- 1} O_{X} \to O_{Z}

. Then the map of schemes

p_{2} \circ ν : {\underline{Z}}^{N} \to \underline{Y}

extends to a map of

\log

schemes

Z^{N} \to Y

Remark 5.6. It follows from (i) and (ii) above that, for

\underline{Z} \in {lor}_{k} (X, Y)

, the restriction of

\underline{Z}

to a cycle on the open subset

X^{\circ} \times Y^{\circ}

\underline{X} \times \underline{Y}

actually lands in

{Cor}_{k} (X^{\circ}, Y^{\circ})

. Moreover, by [19, LemMA 2.3.1], if the extension in (ii) exists, it is unique, so there is no need to include this as part of the data. In particular, the restriction map

{lor}_{k} (X, Y) \to {Cor}_{k} (X^{\circ}, Y^{\circ})

is injective ([19, LEMMA 2.3.2]).

The condition that there exists a map of

\log

schemes

(Z^{N}, {(p_{1} \circ ν)}_{\log}^{*} M_{X}) \to

(Y, M_{Y})

extending

p_{2} \circ v : {\underline{Z}}^{N} \to \underline{Y}

is analogous to the modulus condition

v^{*} (D \times Y) \geq v^{*} (X \times E)

for a subvariety

W \subset X ∖ D \times Y ∖ E

to define a finite correspondence of modulus pairs from

(X, D)

(Y, E)

For the composition law, the proof of [19, LEMMA 2.3.3] shows that, given elementary

\log

correspondences

\underline{W} \in {lor}_{k} (X, Y)

, and

{\underline{W}}^{'} \in 1 {Cor}_{k} (Y, Z)

, each integral component

\underline{R}

\underline{W} \times \underline{Z} \cap \underline{X} \times \underline{W^{'}}

is the underlying scheme of a (unique!) elementary log correspondence

\underline{R} \in {lor}_{k} (X, Z)

. It is then easy to show that there is a unique composition law

\circ : {lCor}_{k} (Y, Z) \times {lor}_{k} (X, Y) \to {lor}_{k} (X, Z)

that is compatible with the composition law in

{C o r}_{k}

via the respective restriction maps.

This defines the additive category of finite log correspondences

{1 C o r}_{k}

with the same objects as for

1 {S m}_{k}

, giving the category of presheaves with log transfers,

{P S S T}_{k}

, defined as the category of additive presheaves of abelian groups on

1 {lor}_{k}

. For a log scheme

X \in 1 {S m}_{k}

, let

Z_{l t r} (X)

denote the representable presheaf

Z_{1 t r} (X) (Y) := {lor}_{k} (Y, X)

The fiber product of log schemes induces a tensor product structure on

{lPST}_{k}

The next step is to define the log version of the Nisnevich topology.

A morphism of log schemes

f : (X, M_{X} \to O_{X}), (Y, M_{Y} \to O_{Y})

is strict if the map of

\log

structures

f^{*} M_{Y} \to M_{X}

is an isomorphism. An elementary log Nisnevich square is a cartesian square in the category of fs log schemes

where

f

is strict Ã©tale,

g

is an open immersion, and

f

induces an isomorphism on reduced schemes

\underline{Y} ∖ \underline{V} \to \underline{X} ∖ \underline{U}

A log modification is a generalization of the notion of a log blow-up, which in turn is a morphism of log schemes modeled on the birational morphism of toric varieties given by a subdivision of the fan defining the target. We refer the reader to [19, APPENDIX A] for details. The Grothendieck topology generated by the log modifications and strict Nisnevich elementary squares is called the dividing Nisnevich topology on fs log schemes. In a sense, this is a log version of the cdh topology, where all the modifications are taking place in the boundary.

With this topology in hand, we have the subcategory

{lNST}_{k}

{IPST}_{k}

of Nisnevich sheaves with log transfers, just as for

{N S T}_{k} \subset {P S T}_{k}

, by requiring that a presheaf with log transfers be a sheaf for the dividing Nisnevich topology when restricted to

{S m}_{k}

Finally, we need a suitable interval object to define a good notion of homotopy invariance. This is just as for the category

{MDM}^{eff} (k)

, where we consider

\bar{◻}

as the scheme

P^{1}

with compactifying log structure for

(P^{1}, {1})

. The product log scheme

{\bar{◻}}^{2}

also has the compactifying log structure for the divisor

1 \times P^{1} + P^{1} \times 1

. However, the closure

{\bar{Γ}}_{m}

of the graph of the multiplication map

m : {\bar{◻}}^{2} \to \bar{◻}

is not a morphism

\tilde{m}

1 {Cor}_{k}

, as the requirement that the map of

{\bar{Γ}}_{m}

{\bar{◻}}^{2}

be finite is not satisfied.

Another way to look at this is to note that the projection

{\bar{Γ}}_{m} \to {\bar{◻}}^{2}

is a cover of

{\bar{◻}}^{2}

in the dividing Nisnevich topology, and becomes an isomorphism after

d

Nis-localization. In a sense, this allows one to consider the sheaf

a_{d N i s} \bar{◻}

as a version of a cylinder object and allows many of the constructions of Morel-Voevodsky for a site with interval to go through, although there are occasional technical difficulties that arise.

Definition 5.7. The tensor triangulated category of effective log motives over

k, logDM^{e f f} (k)

, is the Verdier localization of the derived category

D ({P S S T}_{k})

with respect to the localizing subcategory generated by:

(IMV) for an elementary log Nisnevich square

we have the complex

Z_{l t r} (V) \to Z_{l t r} (U) \oplus Z_{l t r} (Y) \to Z_{l t r} (X)

(IM) For a log modification

f : Y \to X

1 {S m}_{k}

, we have the complex

Z_{l t r} (Y) \to Z_{l t r} (X)

(lCI) For

X \in 1 {S m}_{k}

, we have the complex

Z_{l t r} (X \times ⊵) \to Z_{l t r} (X)

For each fs smooth log scheme

X \in 1 {Sm}_{k}

, the image of

Z_{ltr} (X)

logDM^{eff} (k)

is the effective log motive

{l M}^{eff} (X)

, giving the functor

{l M}^{e f f} : 1 {S m}_{k} \to logDM^{e f f} (k)

The functor

{I M}^{eff}

shares many of the formal properties of

M^{eff} : {Sm}_{k} \to {D M}^{eff} (k)

; we refer the reader to the [19, INTRoDUction] for an overview.

Questions of representing known constructions such as the higher Chow groups with modulus in

\log {D M}^{eff} (k)

, or finding direct connections of

\log {D M}^{eff} (k)

with the category

{MDM}^{eff} (k)

are not discussed in [19]. However, for

(X, D)

a proper modulus pair, one has the log scheme

l (X, D)

, defined using the Deligne-Faltings log structure on

X

associated to the ideal sheaf

O_{X} (- D)

. In general, this is not saturated. Still, there should be presheaves with

\log

transfers

Z_{l t r} (X, D)

and

Z_{l t r}^{c} (X, D)

using finite and quasi-finite "log correspondences," with value on

Y \in 1 {Sm}_{k}

the free abelian group of integral subschemes

W

\underline{Y} \times X

that admit a map of log schemes

(W^{N}, {(p_{1} \circ ν)}^{*} (M_{Y})) \to l (X, D)

, as in the definition of

1 {Cor}_{k} (-, -)

. One could also expect to have presheaves

l z (X, D, r)

similarly defined, and corresponding to the presheaves with modulus transfers

z (X, D, r)

constructed by Miyazaki. These could be used to give a map

H_{p} (z_{r}^{equi} (X, D, *)) \to {Hom}_{\log D M^{eff} (k)} (Z (0) [p], M (l z (X, D, r)))

We have briefly mentioned reciprocity sheaves in our discussion of motives with modulus. There is a nice connection of

logDM^{eff} (k)

with the theory of reciprocity sheaves, so we take the opportunity to say a few words about reciprocity sheaves before we describe the theorem of Shuji Saito, which gives the connection between these two theories.

The notion of a reciprocity sheaf and its relation to motives with modulus goes back to the theorem of Rosenlicht-Serre. In our discussion of reciprocity sheaves, we work over a fixed perfect field

k

Theorem 5.8 (Rosenlicht-Serre

[109

, III]). Let

k

be a perfect field, let

C

be a smooth complete curve over

k

, let

G

be an smooth commutative algebraic group over

k

, and let

f

C \to G

be a rational map over

k

. Let

S \subset C

be a finite subset such that

f

is a morphism on

C ∖ S

. Then there is an effective divisor

D

supported in

S

such that, for

g

a rational function on

C

with

g \equiv 1 mod D

, one has

\sum_{P \in C ∖ S} {ord}_{P} (g) \cdot f (P) = 0

G

In [72], reciprocity functors and reciprocity sheaves are defined. We will just give a sketch. One first defines for

F

a presheaf with transfers (in the Voevodsky sense), and for a proper modulus pair

(X, D)

with a section

a \in F (X ∖ D)

, what it means for

a

to have modulus

D

. As an example, if

p : C \to X

is a non-constant morphism of a smooth proper integral curve

C

over

k

X

with

p (C)

not contained in

D

, and

g

is a rational function on

C

such that

g \equiv 1 mod p^{*} (D)

, then one is required to have

a (p_{*} (div (g))) = 0 \in F (Spec k)

Here, for a 0 -cycle

\sum_{x} n_{x} \cdot x

X ∖ D, a (\sum_{x} n_{x} \cdot x) = \sum_{x} n_{x} \cdot p_{x * i_{x}^{*}} (a) \in F (Spec k)

, where for a closed point

x

X ∖ D, i_{x} : x \to X ∖ D

is the inclusion and

p_{x} : x \to Spec k

is the (finite) structure morphism. In general, one imposes a similar condition in

F (S)

for a "relative curve" on

X \times S

over some smooth base scheme

S

A presheaf with transfers

F

is a reciprocity sheaf if for each quasi-affine

U

and section

a \in F (U)

, there is a proper modulus pair

(X, D)

with

U = X ∖ D

such that

a

has modulus

D

. Roughly speaking, one should think that each section of

F

has "bounded ramification," although the "ramification" for

F

itself may be unbounded.

This definition is not quite accurate, as a slightly different notion of "modulus pair" from what we have defined here is used in [72]. A more elegant definition of reciprocity sheaf is given in [73]. This new notion is a bit more restrictive than the old one, but by [73, THEOREM 2], the two notions agree on for

F \in \underline{{M N S T}_{k}}

Using the definition of [73], the reciprocity sheaves define a the full subcategory

{R S T}_{k}

{P S T}_{k}

, strictly containing the subcategory

{H I}_{k} \subset {P S T}_{k}

A^{1}

-homotopy invariant presheaves with transfer. There is also the subcategory

{R S T}_{N i s, k}

{N S T}_{k}

, consisting of those reciprocity presheaves that are Nisnevich sheaves.

Some examples of non-homotopy invariant sheaves in

{R S T}_{N i s, k}

include the sheaf of

n

-forms over

k, X \mapsto Ω_{X / k}^{n}

, the sheaf of absolute

n

-forms,

X \mapsto Ω_{X}^{n}

, and for

k

of positive characteristic, the truncated de Rham-Witt sheaves,

X \mapsto W_{m} Ω_{X}^{n}

. The representable sheaf of a commutative algebraic group

G

over

k, X \mapsto G (X)

, is in

{R S T}_{Nis, k}

, and for some

G

(e.g.

G = G_{m}^{n}

) this is also

A^{1}

-homotopy invariant. This is not the case for unipotent

G

(e.g.

G = G_{a}^{n})

Here is the promised theorem of Saito. For a sheaf

G \in {NNST}_{k}

, we say that

G

H_{d N i s}^{*} (X, G_{∣ X_{d N i s}}) \to H_{d N i s}^{*} (X \times \bar{◻}, G_{∣ X \times {\bar{◻}}_{d N i s}})

induced by the projection

X \times \bar{◻} \to X

is an isomorphism. Here

d

Nis refers to the divided Nisnevich site.

Theorem 5.9 (Saito [105, THEOREM 0.2]). There exists a fully faithful exact functor

\log : {R S T}_{N i s, k} \to {lNST}_{k}

such that

\log (F)

is strictly

\bar{◻}

-invariantfor every

F \in {R S T}_{N i s, k}

. Moreover, for each

X \in {Sm}_{k}

, there is a natural isomorphism

H_{N i s}^{i} (X, F_{∣ X}) ≅ {Hom}_{{logDM}^{e f f} (k)} ({lM}^{e f f} (X), \log (F) [i])

6. $p$ -ADIC Ã‰TALE MOTIVIC COHOMOLOGY AND $p$ -ADIC HODGE
THEORY

We discuss yet another theory of motivic cohomology that is not

A^{1}

-homotopy invariant.

Working over a base-field

k

and for

m

Z / m (r)_{ét}

μ_{m}^{\otimes r}

Z / m (r)_{ét}

can be considered as objects in a version of Voevodsky's DM constructed using the Ã©tale topology rather than the Nisnevich topology, and their categorical cohomology agrees with the usual Ã©tale cohomology [127, cHAP. 5, Â§3.3]. In particular, the complexes

Z / m (r)_{ét}

have

A^{1}

On the other hand, if

k

has characteristic

p > 0

, we have the isomorphism [52] of the Nisnevich sheaves on

{S m}_{k}

\begin{matrix} (6.1) & Z / p^{n} (r) ≅ W_{n} Ω_{\log}^{r} [- r] \end{matrix}

Z / p^{n} (r)_{é t} ≅ W_{n} Ω_{\log}^{r} [- r]

Z / p^{n} (r)_{ett}

A^{1} \to A^{1}

of degree

p

{D M}^{eff} (k)

with coefficients modulo

p^{n}

is zero if

k

has characteristic

p > 0

[127, CHAP. 5, PROPOSITION 3.3.3]. Thus

Z / p^{n} (r)_{ét}

leaves the world of Voevodsky's motives and motivic cohomology.

For

S = Spec Λ

, with

Λ

a mixed characteristic

(0, p) d v r

, the complex

Z / p^{n} (r)_{et}

{Sm}_{S, ét}

yields an interesting gluing of

Z / p^{n} (r)_{ét} = μ_{p^{n}}^{\otimes r}

over the characteristic zero quotient field of

Λ

and

Z / p^{n} (r)_{et} = W_{n} Ω_{\log}^{r} [- r]

over the characteristic

p

residue field. The positive characteristic part again says that we have left homotopy invariance behind.

The complexes

Z / p^{n} (r)_{ét}

have an interesting connection with a certain complex of sheaves arising in

p

-adic Hodge theory. A version of this complex first appears in the paper [40] of Fontaine-Messing, and plays an important role in the proof of their main result. Its construction was reinterpreted by Kurihara [82], relying on the work of Bloch-Kato [28] and Kato [79], and was generalized by Sato [106]. Geisser [49], following work of Schneider [108], established the connection of the Fontaine-Messing/Kurihara/Sato complex with

Z / p^{n} (r)_{êt}

in the case of a smooth degeneration, and this connection was partially extended by Zhong [128] to the semi-stable case.

In their recent work on integral

p

-adic Hodge theory, Bhatt-Morrow-Scholze [18] have introduced a "motivic filtration" on

p

-adic Ã©tale

K

-theory, relying on a Postnikov tower for topological cyclic homology, and the layers in this tower have been identified with the pro-system

{Z / p^{n} (r)_{ét}}_{n}

in a work-in-progress [16] by Bhargav Bhatt and Akhil Mathew.

Our goal in this section is to give some details of the story sketched above.

We first discuss the papers of Bloch-Kato, Fontaine-Messing, Kurihara and Sato without reference to all the advances in

p

-adic Hodge theory that followed these works; we wanted to give the reader just enough background to put the connections with motivic
cohomology in context. We will then describe the works of Geisser and Zhong, as well as results of Geisser-Hesselholt that form some of the foundations for the work of BhattMorrow-Scholze. We conclude with a description of the Bhatt-Morrow-Scholze motivic tower and its connection with the

p

-adic cycle complexes.

We refer the reader to [15] for background on crystalline cohomology.

6.1. A quick overview of some $p$ -adic Hodge theory

We begin with a few comments on the paper of Bloch and Kato [28], which we have already mentioned in our discussion of the Beilinson-Lichtenbaum conjectures. They consider the spectrum

S

of a complete

dvr Λ

with generic point

η = Spec K ↪ S

and closed point

s = Spec k ↪ S

, and a smooth and proper

S

-scheme

X \to S

with generic fiber

V := X_{η}

and special fiber

Y := X_{s} . \bar{V}, \bar{Y}

denote

V, Y

over the respective algebraic closures

\bar{K}

and

\bar{k}

K

and

k

. Let

\bar{Λ}

be the integral closure of

Λ

\bar{K}, \bar{S} := Spec \bar{Λ}

, and

\bar{X} = X \times_{S} \bar{S}

. Let

G = Gal (\bar{K} / K)

and let

C

denote the completion of

\bar{K}

The closure

\bar{Y}

has its crystalline cohomology

H_{crys}^{*} (\bar{Y} / W (\bar{k}))

with action of Frobenius, giving the

p^{i}

-eigenspace

H_{crys}^{*} (\bar{Y} / W (\bar{k}))^{(i)} \subset H_{crys}^{*} (\bar{Y} / W (\bar{k}))_{Q}

We say

\bar{Y}

is ordinary if

\dim_{W (\bar{k})_{Q}} H_{c r y s}^{m} (\bar{Y} / W (\bar{k}))^{(i)} = \dim_{\bar{k}} H^{m - i} (\bar{Y}, Ω_{\bar{Y} / \bar{k}}^{i})

We have the inclusions

\bar{i} : \bar{Y} \to \bar{X}, \bar{j} : \bar{V} \to \bar{X}

and the spectral sequence

E_{2}^{s, t} = H_{e t}^{s} (\bar{Y}, {\bar{i}}^{*} R^{t} {\bar{j}}_{*} (Z / p^{n} Z)) \Rightarrow H_{e t}^{s + t} (\bar{V}, Z / p^{n} Z)

inducing a descending filtration

F^{*} H_{e t}^{*} (\bar{V}, Q_{p})

H_{e t}^{*} (\bar{V}, Q_{p})

with

F^{0} H^{q} = H^{q}

and

F^{q + 1} H^{q} = 0

We have the de Rham-Witt sheaf

W Ω^{i}

{S m}_{\bar{k}}

and the sheaf of differential forms

Ω_{- / K}^{i}

{S m}_{K}

Theorem 6.1 (Bloch-Kato [28, Ñ‚HEORem 0.7]). Suppose that

k

is perfect and that

Y

is ordinary. Then there are natural

G

-equivariant isomorphisms

(i)

{gr}^{q - i} H_{e t}^{q} (\bar{V}, Q_{p}) ≅ H_{c r y s}^{q} (\bar{Y} / W (\bar{k}))_{Q}^{(i)} (- i)

(ii)

{gr}^{q - i} H_{e t}^{q} (\bar{V}, Q_{p}) \otimes_{Z_{p}} W (\bar{k}) ≅ H_{crys}^{q} {(\bar{Y}, W Ω^{i})}_{Q} (- i)

(iii)

{g r}^{q - i} H_{e t}^{q} (\bar{V}, Q_{p}) \otimes_{Q_{p}} C ≅ H^{q} (V, Ω_{V / K}^{i}) \otimes_{K} C (- i)

We will not give any details of the proof here, but do want to mention that what ties these different theories together is the sheaf of Milnor

K

-groups

K_{q}^{M}

θ_{q, m} : K_{q}^{M} / m \to H_{e t}^{q} (μ_{m}^{\otimes q})

for

m

prime to the characteristic, to the de Rham-Witt sheaf by the

d \log

map on

{S m}_{k}

d \log_{q, p^{n}}^{W} : K_{q}^{M} / p^{n} \to W_{n} Ω^{q}

and to the sheaf of differential forms, by the

d \log

map on

{S m}_{S}

d \log_{q, p^{n}} : K_{q}^{M} / p^{n} \to Ω_{- / S}^{q} / p^{n}

The main structural results that underpin the proof of the Bloch-Kato theorem are two comparison isomorphisms on the sheaf level. For the first, let

W_{n} Ω_{\log}^{q} \subset W_{n} Ω^{q}

d \log (K_{q}^{M} / p^{n})

Theorem 6.2 ([28, COROLLARY 2.8]). Let

F

be a field of characteristic

p > 0

. Then the map

d \log : K_{q}^{M} (F) / p^{n} \to W_{n} Ω^{q} (F)

defines an isomorphism of

K_{q}^{M} (F) / p^{n}

with

W_{n} Ω_{\log}^{q} (F)

Note that the composition

Z / p^{n} (q) \to τ_{\geq q} Z / p^{n} (q) ≅ K_{q}^{M} / p^{n} [- q] \overset{d \log}{\to} W_{n} Ω_{\log}^{q} [- q]

is the map that defines the isomorphism (6.1).

The second result is a special case of the Bloch-Kato conjecture.

Theorem 6.3 (Bloch-Kato [28, THEOREM 5.12]). Let F be a henselian discretely valued field of characteristic 0 , with residue field of characteristic

p > 0

. Then the Galois symbol

K_{q}^{M} (F) / p^{n} \to H_{e t}^{q} (F, μ_{p^{n}}^{\otimes q})

is an isomorphism for all

n \geq 1 .^{2}

Bloch and Kato use

K_{q}^{M}

to relate

i^{*} R^{q} j_{*} μ_{p^{n}}^{\otimes q}

Ω_{- / K}^{q} / p^{n}

and

W_{n} Ω_{\log}^{q - 1}

via the respective

d \log

maps. Relying on the isomorphisms of Theorem 6.2 and Theorem 6.3, these maps from Milnor

K

-theory tie de Rham cohomology, crystalline cohomology and Ã©tale cohomology together, and eventually lead to a proof of Theorem 6.1.

As part of the proof, they define a surjective map

\begin{matrix} (6.2) & γ : i^{*} R^{q} j_{*} μ_{p^{n}}^{\otimes r} \to W_{n} Ω_{\log}^{q - 1} \end{matrix}

Y_{ét}

with the following property: Let

θ : i^{*} j_{*} K_{q, ét}^{M} \to i^{*} R^{q} j_{*} μ_{p^{n}}^{\otimes r}

be the Galois symbol map, let

u_{2}, \dots, u_{q}

be units on

X

near some point

y

Y

with restrictions

{\bar{u}}_{1}, \dots, {\bar{u}}_{q}

Y

and let

π

be a parameter in

Λ

. Then

γ \circ θ ({u_{1}, \dots, u_{q - 1}, π}) = d \log ({{\bar{u}}_{1}, \dots, {\bar{u}}_{q - 1}})

We highlight this because it will be used later on in a gluing construction that defines an object of central interest for this section.

The next paper I want to mention is by Fontaine-Messing [40]. They construct a comparison isomorphism between de Rham cohomology and Ã©tale cohomology for a smooth, proper scheme

X

over the ring of integers

O_{K}

for

K

a characteristic zero local field (under some additional assumptions). The de Rham cohomology

H_{d R}^{q} (V / K)

has its Hodge filtration and via the comparison isomorphism

H_{d R}^{q} (V / K) ≅ H_{c r y s}^{q} (Y / W (k)) \otimes_{W (k)} K

H_{d R}^{q} (V / K)

acquires a Frobenius operator

ϕ

; call this object

H_{crys}^{q} (X)

. Fontaine-Messing construct the

p

-adic period ring

B_{crys} \supset K

with a Galois action, a Frobenius and a filtration, and show there are isomorphisms

{Fil}^{0} {(B_{crys} \otimes_{K} H_{crys}^{q} (X))}^{ϕ = I d} ≅ H_{e t}^{q} (V_{\bar{K}}, Q_{p})

and

{(B_{crys} \otimes_{Q_{p}} H_{e t}^{q} (V_{\bar{K}}, Q_{p}))}^{G} ≅ H_{crys}^{q} (X)

both compatible with the "remaining" structures.

To set this up, they consider the syntomic topology on

{S c h}_{W_{n} (k)}

, where a cover is a surjective syntomic map (we described syntomic maps in Section 3.3). The crystalline structure sheaf

O_{n}^{crys}

defines a sheaf for the syntomic topology with a surjection to the usual structure sheaf

O_{n}

{Sch}_{W_{n} (k)}

. Letting

J_{n}

denote the kernel of

O_{n}^{crys} \to O_{n}

, one has the

r

th divided power

J_{n}^{[r]}

; this gives us the sheaf

{\tilde{S}}_{n}^{r}

defined as the kernel of

ϕ - p^{r} : J_{n}^{[r]} \to O_{n}^{crys}

. Modifying this by taking the image

S_{n}^{r}

of the reduction map

{\tilde{S}}_{n + r}^{r} \to {\tilde{S}}_{n}^{r}

gives the inverse system

{S_{n}^{r}}_{n}

and the cohomology

H^{*} (\bar{X}, S_{Q_{p}}^{r}) := (lim_{\overset{\leftarrow}{n}} H^{*} (\bar{X}, S_{n}^{r})) \otimes_{Z_{p}} Q_{p}

The ring

B_{crys}^{+}

is defined as follows. The characteristic

p

ring

O_{\bar{K}} / p

forms an inverse system via the Frobenius endomorphism; let

O^{b} = \lim_{Frob} O_{\bar{K}} / p

a perfect characteristic

p

ring. We have the ring of truncated Witt vectors

W_{n} (O^{b})

and a surjection

π_{n} : W_{n} (O^{b}) \to O_{\bar{K}} / p^{n}

. Let

W_{n}^{D P} (O_{\bar{K}})

be the divided power envelope of the kernel of

π_{n}

, forming the inverse system

{W_{n}^{D P} (O_{\bar{K}})}_{n \geq 0}

. Let

B_{crys}^{+} := K \otimes_{W (k)} lim_{\overset{\leftarrow}{n}} W_{n}^{D P} (O_{\bar{K}})

The Frobenius on

W_{n} (O^{b})

induces a Frobenius on

B_{crys}^{+}

and the filtration

J_{n}^{[*]}

W_{n}^{D P} (O_{\bar{K}})

induces a filtration Fil*

B_{crys}^{+}

B_{crys}^{+}

The derived push-forward of the complex

J_{n}^{[r]} \overset{ϕ - p^{r}}{\to} O_{n}^{crys}

is an analog of the Deligne complex, as expressed in the following theorem.

Theorem 6.4 ([40, corollary to theorem 1.6, LemMA 3.1]). Suppose that

X

is admissible

^{3}

and

Λ = W (k)

. Then for

m \leq r < p

there is an exact sequence

0 \to H^{m} (\bar{X}, S_{Q_{p}}^{r}) \to {Fil}^{r} (B_{crys}^{+} \otimes_{K} H_{d R}^{m} (V / K)) \overset{ϕ - p^{r}}{\to} H_{d R} (V / K) \to 0

In other words,

H^{m} (\bar{X}, S_{Q_{p}}^{r}) = {({Fil}^{r} (B_{crys}^{+} \otimes_{K} H_{d R}^{m} (V / K)))}^{ϕ = p^{r}}

3 See

[40, $ 2.1]

for the definition of admissible

X

Spf (W (k))

-schemes, where a cover is a map

{U_{n}}_{n} \to {V_{n}}_{n}

such that

U_{n} \to V_{n}

is a syntomic cover for all

n

\bar{X}

, where an object is

U \to \bar{X}

, quasi-finite and syntomic, with

U_{\bar{K}} \to V_{\bar{K}}

Z

be the formal completion of

\bar{X}

, we have the diagram of sites

Z_{syn-ét} \overset{i}{\to} {\bar{X}}_{syn-ét} \overset{j}{\leftarrow} V_{\bar{K}, ét}

Fontaine-Messing prove a patching result, that a sheaf on

{\bar{X}}_{syn-ét}

is given by a triple

(F, E, α)

, with

F

a sheaf on

Z_{syn-ét}, E

a sheaf on

V_{\bar{K}, e t}

, and

α : F \to i^{*} R j_{*} E

a morphism. Using this description, they construct a sheaf on

{\bar{X}}_{syn-ét}

by defining a certain morphism (see

[40, § 5.1])

α : S_{n}^{r} \to i^{*} R j_{*} μ_{p^{n}}^{\otimes r}

The resulting sheaf

{\tilde{S}}_{n}^{r}

has

j^{*} {\tilde{S}}_{n}^{r} ≅ μ_{p^{n}}^{\otimes r}, i^{*} {\tilde{S}}_{n}^{r} ≅ S_{n}^{r}

It follows from proper base-change (see the proof of [40, PROPOSITION 6.2]) that the restriction map

H^{*} ({\bar{X}}_{syn-é}, {\tilde{S}}_{n}^{r}) \to H^{*} (Z_{syn-et}, S_{n}^{r})

is an isomorphism, and we also have

(lim_{\overset{\leftarrow}{n}} H^{*} (Z_{s y n - e t}, S_{n}^{r})) \otimes_{Z_{p}} Q_{p} ≅ {({Fil}^{r} (B_{c r y s}^{+} \otimes_{K} H_{d R}^{m} (V / K)))}^{ϕ = p^{r}}

Moreover, the restriction map

j^{*}

gives

j^{*} : H^{*} ({\bar{X}}_{s y n - e t}, {\tilde{S}}_{n}^{r}) \to H_{e t}^{*} (V_{\bar{K}}, μ_{p^{n}}^{\otimes r})

so passing to the limit, we have the map

β : {({Fil}^{r} (B_{crys}^{+} \otimes_{K} H_{d R}^{m} (V / K)))}^{ϕ = p^{r}} \to H_{e t}^{m} (V_{\bar{K}}, Q_{p} (r))

which they show is an isomorphism.

This gives a twisted version of the result announced at the beginning of our discussion. To recover the untwisted version, they define a map

Q_{p} (1) (\bar{K}) \to B_{crys}^{+}

by sending a

p^{n}

-root of unity

ε

\bar{K}

to the logarithm of the TeichmÃ¼ller lift of the mod

p

reduction of

ε

, and passing to the limit in

n

. Let

t \in Q_{p} (1) (\bar{K})

be a nonzero element and define

B_{crys} = B_{crys}^{+} [1 / t]

, with induced filtration and Galois action. Twisting with respect to

t

translates the twisted version to the untwisted one.

The sheaf

{\tilde{S}}_{n}^{r}

is only defined on

{\bar{X}}_{syn-ét}

for

X

smooth over

S

and for

r < p

, and with base-ring

Λ

equal to

W (k)

, i.e., in the unramified case. Kato [79] studies the derived push-forward

S_{n} (r)

of the syntomic sheaf

{\tilde{S}}_{n}^{r}

{S m}_{k, et}

. Kurihara [82,

§ 1

, THEOREM] considers the ramified case and also clarifies the relation of

S_{n} (r)

with the sheaf of

\log

forms

W_{n} Ω_{\log}^{r - 1}

Theorem 6.5 (Kurihara). Suppose that

[k : k^{p}] < \infty

. Let

X \to S

be smooth and projective and suppose that

r < p - 1

. Then there is a distinguished triangle in

D (Y_{\hat{e t}})

W_{n} Ω_{\log}^{r - 1} [- r - 1] \to S_{n} (r) \to i^{*} R j_{*} μ_{p^{n}}^{\otimes r} \to W_{n} Ω_{\log}^{r - 1} [- r]

Schneider [108] extends the construction of

S_{n} (r)

to all

r \geq 0

by using the BlochKato symbol map

γ

of (6.2) to give a map

s : τ_{\leq r} R j_{*} μ_{p^{n}}^{\otimes r} \to i_{*} W_{n} Ω_{\log}^{r - 1}

with

i^{*} s

the composition

i^{*} τ_{\leq r} R j_{*} μ_{p^{n}}^{\otimes r} \to i^{*} R^{r} j_{*} μ_{p^{n}}^{\otimes r} [- r] \overset{γ}{\to} W_{n} Ω_{\log}^{r - 1} [- r] .

Schneider then defines the sheaf

S_{n} (r)

as Cone

(s) [- 1]

, giving the distinguished triangle

\begin{matrix} (6.3) & i_{*} W_{n} Ω_{\log}^{r - 1} [- r - 1] \to S_{n} (r) \to R j_{*} μ_{p^{n}}^{\otimes r} \overset{s}{\to} i_{*} W_{n} Ω_{\log}^{r - 1} [- r], \end{matrix}

which recovers the one in Kurihara's theorem for

r < p - 1

by applying

i^{*}

Using a similar method, Schneider's construction was extended to the semi-stable case by Sato [106], who defines the object

T_{n} (r) \in D (X_{et})

with

T_{n} (r) ≅ S_{n} (r)

in the smooth case.

6.2. Ã‰tale motivic cohomology

We return to algebraic cycles. As before, we consider a smooth separated finite type

S

-scheme

X \to S = Spec Λ

with generic fiber

j : V \to X

and special fiber

i : Y \to X

, and with

Λ

a mixed characteristic

(0, p)

dvr with perfect residue field.

Geisser [49] considers the motivic complex

Z (r)_{X}

on a smooth

S

-scheme

X \to S

as a sheaf of complexes on

X_{N i s}

. Here we use the reindexed Bloch cycle complex to define

Z (r)_{X}^{*} (U)

Z (r)_{X}^{*} (U) := z^{r} (U, 2 r - *)

and define the motivic complexes

Z (r)_{V}

and

Z (r)_{Y}

V

and

Y

similarly.

Let

α : (-)_{êt} \to (-)_{Nis}

Z (r)_{e t, X}, Z (r)_{e t, V}

, and

Z (r)_{e t, Y}

. Geisser shows that various known properties of

Z (r)_{X}, Z (r)_{V}

, and

Z (r)_{Y}

, such as the purity isomorphism [84, THEOREM 1.7]

i^{!} Z (r)_{X} ≅ Z (r - 1)_{Y} [- 2]

the theorem of Geisser-Levine [52]

Z / p^{n} (r)_{Y} ≅ W_{n} Ω_{\log, Y}^{r} [- r]

the Suslin-Voevodsky isomorphism in

D^{b} (V_{êt})

(Beilinson's axiom (iv)(a))

j^{*} Z / p^{n} (r)_{e t, X} ≅ Z / p^{n} (n)_{e t, V} ≅ μ_{p^{n}}^{\otimes r}

and the Beilinson-Lichtenbaum conjectures (now a theorem)

Z / p^{n} (r)_{V} ≅ τ_{\leq r} R α_{*} μ_{p^{n}}^{\otimes r}, R^{r + 1} α_{*} Z (r)_{e t, V} = 0

have as consequence

Theorem 6.6 (Geisser [49, THEOREM 1.3]). Let

X \to

Spec

Λ

be smooth and essentially of finite type, with

Λ

a complete discrete valuation ring of mixed characteristic

(0, p)

. Then there is a distinguished triangle in

D^{b} (X_{èt})

i_{*} W_{n} Ω_{\log}^{r - 1} [- r - 1] \to Z / p^{n} (r)_{e ́ t} \to τ_{\leq r} R j_{*} μ_{p^{n}}^{\otimes r} \to i_{*} W_{n} Ω_{\log}^{r - 1} [- r]

and an isomorphism

Z / p^{n} (r)_{e ́ t} ≅ S_{n} (r)

D^{b} (X_{\dot{e} t})

that transforms this triangle to Schneider's defining triangle (6.3).

Zhong has extended this to the semi-stable case, establishing an isomorphism with Sato's construction

I_{n} (r)

after a truncation [128, PROPOSITION 4.5]:

τ_{\leq r} Z / p^{n} (r)_{ét} ≅ T_{n} (r)

Assuming a "weak Gersten conjecture" for

Z / p^{n} (r)_{êt}

, the truncation is removed [128, tHEoREM 4.8].

6.3. The theorems of Geisser-Hesselholt

The construction of a motivic tower for integral

p

-adic Hodge theory by BhattMorrow-Scholze relies on properties of

p

-completed topological cyclic homology, including the results of Geisser-Hesselholt identifying this with the

p

K

dvr Λ

with perfect residue field

k

Topological cyclic homology for a fixed prime

p

is a spectrum refined version of Connes' cyclic homology and is defined for a scheme

X

with a topology

τ \in {

{{TC}^{m} (X, p)}_{m \in N}

defining

T C (X; p)

as the homotopy inverse limit

T C (X; p) := \underset{m}{holim} {T C}^{m} (X, p)

Let

T C_{i}

i

th pro-homotopy groups

U \mapsto π_{i} T C \cdot (U; p)

. There is a descent spectral sequence

E_{2}^{s, t} = H_{c o n t}^{s} (X, T e_{- t}^{\cdot}) \Rightarrow {T C}_{- s - t} (X; p)

and a cyclotomic trace map

trc : K (X) \to T C (X; p)

Let

k

be a perfect field of characteristic

p > 0

. It follows from a result of Hesselholt [60, THEOREM B] that there is a isomorphism of pro-sheaves on

{S m}_{k ét}

\begin{matrix} (6.4) & T C_{r} ≅ W \cdot Ω_{\log}^{r} \end{matrix}

The map trc induces the map of pro-sheaves on

{S m}_{k ét}

trc : K_{i} (Z / p^{\cdot}) \to T C_{i}

where

K_{i} (Z / p^{\cdot})

U \mapsto

{K_{i} (U, Z / p^{ν})}_{ν}

. Relying on the main theorem of [52] and the isomorphism (6.4), Geisser and Hesselholt show

Theorem 6.7 (Geisser-Hesselholt [50, COROLLARY 4.2.5, THEOREM 4.2.6]).

The trace map trc : $K_{i} (Z / p^{\cdot}) \to T Y_{i}$ is an isomorphism of pro-sheaves on ${S m}_{k e ́ t}$ .

K^{e ́ t} (Y)^{\land_{p}} := \underset{m}{holim} K^{e ́ t} (Y, Z / p^{n}) ≅ TC (Y; p)

and this weak equivalence arises from the weak equivalences at the finite level

trc : K^{e t} (Y, Z / p^{ν}) \tilde{\to} T C (Y; p, Z / p^{ν})

Now consider a smooth finite type scheme

X \to

Spec

Λ

with special fiber

i : Y \to X

and generic fiber

j : V \to X

, as before.

Theorem 6.8 (Geisser-Hesselholt [51, THEorems A and B]). Suppose

Λ

is henselian.

A. Suppose

X \to Spec Λ

is smooth and proper. Then

trc : K_{q}^{e t} (X, Z / p^{ν}) \to {T C}_{q} (X; p, Z / p^{ν})

is an isomorphism for all

q \in Z

and

v \geq 1

B. Suppose that

X \to Spec Λ

is smooth and finite type. Then the map of prosheaves on

Y_{e ́ t}

i^{*} K_{q} (Z / p^{ν}) \to {i^{*} T C_{q}^{m} (p, Z / p^{ν})}_{m \in N}

is an isomorphism for all

q \in Z

and all

ν \geq 1

Remark 6.9. To pass from the isomorphism of Theorem 6.7 to that of Theorem 6.8(B), Geisser-Hesselholt rely on the theorem of McCarthy [88], stating that the cyclotomic trace map from relative

K

-theory to relative TC,

trc : K_{q} (X / π^{n}, X / π^{n - r}, Z / p^{ν}) \to {T C}_{q} (X / π^{n}, X / π^{n - r}, Z / p^{ν})

is an isomorphism for affine

X

. Thus, the

K

-theory and topological cyclic homology of non-reduced schemes play a central role in the proof of Theorem 6.8.

6.4. Integral $p$ -adic Hodge theory and the motivic filtration

Bhatt-Morrow-Scholze [17,18] have constructed integral versions of

p

-adic Hodge theory. Here we discuss some aspects of the theory of [18] and its relation to

p

p

-completed) topological Hochschild homology

THH (-, Z_{p})

, topological negative cyclic homology

{TC}^{-} (-, Z_{p})

, and topological periodic cyclic homology

T P (-, Z_{p})

. For a nice, quick overview of these theories, we refer the reader to

[18, $ 1.2, $ 2.3]

, and to

[18

, THEOREM 1.12] for their relation to

T C (-, Z_{p})

Let

C_{p}

be the completion of the algebraic closure of

Q_{p}

, with ring of integers

O_{C_{p}}

. As in our review of the work of Fontaine-Messing, we have the

F_{p}

-algebra

O_{C_{p}} / p

, its perfection

O_{C_{p}}^{b}

and the ring of Witt vectors

A_{inf} (O_{C_{p}}) := W (O_{C_{p}}^{b})

. Hesselholt has connected this with negative cyclic homology

{T C}^{-}

, constructing an isomorphism

π_{0} {T C}^{-} (O_{C_{p}}, Z_{p}) ≅ A_{i n f} (O_{C_{p}})

This has been generalized by Bhatt-Morrow-Scholze in the setting of perfectoid rings (see [17, DEFINITION 3.5]). For a perfectoid ring

R

, we have Scholze's ring

R^{b}

, defined as for

O_{C_{p}}^{b}

by taking the perfection of

R / p

This gives the ring of Witt vectors

A_{inf} (R) := W (R^{b})

with Frobenius

ϕ

induced by the Frobenius on

R^{b}

Theorem 6.10 (Bhatt-Morrow-Scholze [18, THEOREM 1.6]). Let

R

be a perfectoid ring. Then there is a canonical

ϕ

-equivariant isomorphism

π_{0} {T C}^{-} (R, Z_{p}) ≅ A_{inf} (R)

Fix a discretely valued extension

K

Q_{p}

, with ring of integers

O_{K}

having perfect residue field

k

. Let

C

be the completed algebraic closure of

K

, with ring of integers

O_{C}

. Let

A_{inf} := A_{inf} (O_{C})

Let

X

be a smooth formal scheme over

O_{C}

. In [17], Bhatt-Morrow-Scholze construct a presheaf of complexes of

A_{i n f}

-algebras on

X_{Z a r}, A Ω_{X}

, whose Zariski hypercohomology specializes to crystalline cohomology,

p

A_{inf}

, replacing the ring homomorphism

A_{inf} (O_{C_{p}}) \to B_{crys}

used in the Fontaine-Messing theory. In [18], they refine and reinterpret this theory using

{T C}^{-}

. They define the notion of a quasisyntomic ring and the associated quasi-syntomic site [18, DEFINITION 1.7]; this gives the presheaf

π_{0} {T C}^{-} (-; Z_{p})

on the quasi-syntomic site

{q s y n}_{\tilde{A}}

over a quasi-syntomic ring

\tilde{A}

and the associated derived global sections functor

R Γ_{syn} (\tilde{A}, -)

Theorem 6.11 ([18, THEOREM 1.8]). Let

\tilde{A}

be an

O_{C}

-algebra that can be written as the

p

-adic completion of a smooth

O_{C}

-algebra. There is a functorial (in

\tilde{A}

)

ϕ

-equivariant isomorphism of

E_{\infty} - A_{inf}

-algebras

A Ω_{\tilde{A}} ≅ R Γ_{s y n} (\tilde{A}, π_{0} {T C}^{-} (-; Z_{p}))

The Postnikov tower

τ_{\geq *} T C (-; Z_{p})

for the presheaf of spectra

T C (-; Z_{p})

{\tilde{A}}_{qsyn}

induces the tower over

TC (\tilde{A}; Z_{p}

\begin{matrix} (6.5) & \dots \to {Fil}^{n + 1} T C (\tilde{A}; Z_{p}) \to {Fil}^{n} T C (\tilde{A}; Z_{p}) \to \dots \to TC (\tilde{A}; Z_{p} \end{matrix}

by setting

{Fil}^{n} T C (\tilde{A}; Z_{p}) := R Γ_{syn} (\tilde{A}, τ_{\geq 2 n} T C (-; Z_{p}))

(see

[18, $ 1.4])

. Define the sheaves

Z_{p}^{B M S} (n)

by sheafifying the presheaf

\tilde{A} \mapsto Z_{p}^{B M S} (n) (\tilde{A}) := {gr}_{F i l}^{n} T C (\tilde{A}; Z_{p}) [- 2 n]

where

{gr}_{F i l}^{n} T C (\tilde{A}; Z_{p})

is the homotopy cofiber of

{Fil}^{n + 1} T C (\tilde{A}; Z_{p}) \to {Fil}^{n} T C (\tilde{A}; Z_{p})

Theorem 6.12 (

[18

, THEOREM 1.15]).

(1) Let

k

be a perfect field of characteristic

p > 0

, let

A

be a smooth

k

-algebra and let

X = Spec A

X

Z_{p}^{B M S} (r)_{X} ≅ W Ω_{X, \log}^{r} [- r]

(2) Let

C

be an algebraically closed complete extension of

Q_{p}

, let

A

be the completion of a smooth

O_{C}

-algebra, and let

X = Spf A

X

Z_{p}^{B M S} (r) X ≅ τ_{\leq r} R ψ Z_{p} (r)_{X, e ́ t}

Here

Z_{p} (r)_{X, ét}

3.2. Spitzweck's motivic cohomology

3.3. Hoyois' motivic cohomology

4. MILNOR-WITT MOTIVIC COHOMOLOGY

4.1. Milnor-Witt K K KKK-theory and the Chow-Witt groups

Generators

Relations

4.2. The homotopy t t ttt-structure and Morel's theorem

4.3. Milnor-Witt motivic cohomology

4.4. Milnor-Witt motives

5. CHOW GROUPS AND MOTIVIC COHOMOLOGY WITH MODULUS

5.1. Higher Chow groups with modulus

5.2. 0 -cycles with modulus and class field theory

5.3. Categories of motives with modulus

5.4. Logarithmic motives and reciprocity sheaves

Theorem 5.9 (Saito [105, THEOREM 0.2]). There exists a fully faithful exact functor

6. p p ppp-ADIC Ã‰TALE MOTIVIC COHOMOLOGY AND p p ppp-ADIC HODGE THEORY

6.1. A quick overview of some p p p\boldsymbol{p}p-adic Hodge theory

6.2. Ã‰tale motivic cohomology

6.3. The theorems of Geisser-Hesselholt

6.4. Integral p p ppp-adic Hodge theory and the motivic filtration

4.1. Milnor-Witt $K$ -theory and the Chow-Witt groups

4.2. The homotopy $t$ -structure and Morel's theorem

6. $p$ -ADIC Ã‰TALE MOTIVIC COHOMOLOGY AND $p$ -ADIC HODGE
THEORY

6.1. A quick overview of some $p$ -adic Hodge theory

6.4. Integral $p$ -adic Hodge theory and the motivic filtration